💡First, I need to identify the first term and the common difference of the sequence. The first term ( a_1 ) is given as -48. To find the common difference ( d ), I subtract the first term from the second term: ( d = -40 - (-48) = 8 ).Next, I'll use the formula for the nth term of an arithmetic sequence, which is ( a_n = a_1 + (n-1)d ). Plugging in the known values, the formula becomes ( a_n = -48 + (n-1) cdot 8 ).To determine how many terms are in the sequence up to 80, I'll set the nth term equal to 80 and solve for ( n ):[-48 + (n-1) cdot 8 = 80][(n-1) cdot 8 = 80 + 48][(n-1) cdot 8 = 128][n-1 = 16][n = 17]Therefore, there are 17 numbers in the sequence.

💡Alright, so I have this problem about a function ( g ) that's defined for all real numbers. It satisfies two symmetries: ( g(3 + x) = g(3 - x) ) for all ( x ), and ( g(8 + x) = g(8 - x) ) for all ( x ). Also, it's given that ( g(0) = 0 ). The question is asking for the least number of roots ( g(x) = 0 ) must have in the interval ( -1500 leq x leq 1500 ).Hmm, okay. So, first, I need to understand what these symmetries mean. The first equation, ( g(3 + x) = g(3 - x) ), suggests that the function ( g ) is symmetric about the line ( x = 3 ). Similarly, the second equation, ( g(8 + x) = g(8 - x) ), means that ( g ) is also symmetric about ( x = 8 ). So, the function has two lines of symmetry at ( x = 3 ) and ( x = 8 ).Now, if a function is symmetric about two different vertical lines, that usually implies some periodicity. Because if you have two lines of symmetry, the distance between them can be used to find the period. Let me recall that if a function is symmetric about two vertical lines ( x = a ) and ( x = b ), then it's periodic with period ( 2|b - a| ). So, in this case, the distance between 3 and 8 is 5, so the period should be ( 2 times 5 = 10 ). Wait, is that right?Wait, let me think again. If you have two lines of symmetry, the function is periodic with period twice the distance between the two lines. So, the distance between 3 and 8 is 5, so the period is ( 2 times 5 = 10 ). So, ( g(x + 10) = g(x) ) for all ( x ). That seems correct.But let me verify that. Suppose ( g ) is symmetric about ( x = 3 ) and ( x = 8 ). So, reflecting over ( x = 3 ) gives ( g(3 + x) = g(3 - x) ). Similarly, reflecting over ( x = 8 ) gives ( g(8 + x) = g(8 - x) ). So, if I take a point ( x ), reflect it over 3 to get ( 6 - x ), then reflect that over 8, what do I get?Reflecting ( 6 - x ) over 8 would be ( 16 - (6 - x) = 10 + x ). So, ( g(x) = g(6 - x) = g(16 - (6 - x)) = g(10 + x) ). So, indeed, ( g(x) = g(x + 10) ). Therefore, the function is periodic with period 10.Okay, so ( g ) is periodic with period 10. That's a crucial piece of information. So, the function repeats every 10 units. Now, since ( g(0) = 0 ), we can use the periodicity to find other roots.But before that, let's see if there are more roots due to the symmetries. Since ( g ) is symmetric about ( x = 3 ), if ( x ) is a root, then ( 6 - x ) is also a root. Similarly, since it's symmetric about ( x = 8 ), if ( x ) is a root, then ( 16 - x ) is also a root.Wait, but we already have periodicity, so maybe these symmetries are already captured in the periodicity. Let me think.Given that ( g ) is periodic with period 10, and symmetric about ( x = 3 ), then it's also symmetric about ( x = 3 + 10k ) for any integer ( k ). Similarly, it's symmetric about ( x = 8 + 10k ). So, the roots will be symmetric around these points as well.But perhaps it's simpler to consider the periodicity first. Since ( g ) is periodic with period 10, and ( g(0) = 0 ), then ( g(10) = 0 ), ( g(20) = 0 ), and so on. Similarly, ( g(-10) = 0 ), ( g(-20) = 0 ), etc.But wait, is that all? Because we also have the symmetries about 3 and 8. So, if ( g(0) = 0 ), then reflecting over 3 gives ( g(6) = 0 ). Similarly, reflecting over 8 gives ( g(16) = 0 ). But since the function is periodic with period 10, ( g(16) = g(16 - 10) = g(6) = 0 ), which is consistent.So, from ( g(0) = 0 ), we get ( g(6) = 0 ) due to symmetry about 3, and then due to periodicity, ( g(6 + 10k) = 0 ) for all integers ( k ). Similarly, reflecting ( x = 6 ) over 8 gives ( g(10) = 0 ), but ( g(10) = g(0) = 0 ), which is already known.Wait, so perhaps the roots are at ( x = 0 + 10k ) and ( x = 6 + 10k ) for all integers ( k ). So, every 10 units, there are two roots: one at the multiple of 10, and one at 6 more than that multiple.So, in each period of 10, there are two roots. Therefore, in the interval from, say, 0 to 10, we have roots at 0 and 6. From 10 to 20, roots at 10 and 16, and so on.Similarly, on the negative side, from -10 to 0, roots at -10 and -4, and so on.So, in the interval ( -1500 leq x leq 1500 ), how many such roots are there?First, let's find how many periods of 10 are in 3000 units (from -1500 to 1500). Since each period is 10, the number of periods is ( 3000 / 10 = 300 ). But since we're including both endpoints, we need to be careful.Wait, actually, the interval from -1500 to 1500 is 3000 units long. Each period is 10 units, so the number of periods is 300. But in each period, there are two roots. So, the total number of roots would be ( 300 times 2 = 600 ). But wait, we have to check if the endpoints are included or not.Wait, actually, the interval is from -1500 to 1500, inclusive. So, let's think about how many roots are there.First, let's consider the positive side from 0 to 1500. The roots are at 0, 6, 10, 16, 20, 26, ..., up to 1500.Similarly, on the negative side from -1500 to 0, the roots are at -1500, -1494, -1490, -1484, ..., up to 0.But wait, let's think about how many roots are there in total.Each 10 units, there are two roots: one at the multiple of 10, and one at 6 more than that multiple.So, from 0 to 1500, how many multiples of 10 are there? Well, 1500 / 10 = 150. So, 150 roots at multiples of 10, and 150 roots at 6 + multiples of 10. So, total 300 roots from 0 to 1500.Similarly, from -1500 to 0, we have 150 roots at multiples of 10 (including 0) and 150 roots at 6 + multiples of 10 (including 6, but 6 is positive, so on the negative side, it would be -4, -14, etc.). Wait, actually, on the negative side, the roots would be at -10, -16, -20, -26, etc.Wait, let me clarify.The roots are at ( x = 10k ) and ( x = 10k + 6 ) for all integers ( k ).So, for ( k ) such that ( 10k ) is within -1500 to 1500, and ( 10k + 6 ) is also within that interval.So, let's find the range of ( k ) for which ( 10k ) is between -1500 and 1500.So, ( -1500 leq 10k leq 1500 ) implies ( -150 leq k leq 150 ). So, ( k ) ranges from -150 to 150, inclusive. That's 301 values of ( k ) (from -150 to 150 inclusive).Similarly, for ( x = 10k + 6 ), we need ( -1500 leq 10k + 6 leq 1500 ).So, subtracting 6: ( -1506 leq 10k leq 1494 ).Dividing by 10: ( -150.6 leq k leq 149.4 ).Since ( k ) must be an integer, ( k ) ranges from -150 to 149, inclusive. That's 300 values of ( k ).Wait, so for ( x = 10k ), ( k ) ranges from -150 to 150, giving 301 roots.For ( x = 10k + 6 ), ( k ) ranges from -150 to 149, giving 300 roots.So, total roots would be 301 + 300 = 601.But wait, is that correct? Let me check.At ( k = -150 ), ( x = 10(-150) = -1500 ), which is included.At ( k = 150 ), ( x = 10(150) = 1500 ), which is included.For ( x = 10k + 6 ), when ( k = -150 ), ( x = -1500 + 6 = -1494 ), which is within the interval.When ( k = 149 ), ( x = 1490 + 6 = 1496 ), which is within the interval.If ( k = 150 ), ( x = 1500 + 6 = 1506 ), which is outside the interval, so we stop at ( k = 149 ).Similarly, if ( k = -151 ), ( x = -1510 + 6 = -1504 ), which is outside the interval, so we start at ( k = -150 ).So, yes, ( k ) ranges from -150 to 149 for ( x = 10k + 6 ), giving 300 roots.Therefore, total roots are 301 (from multiples of 10) + 300 (from 10k + 6) = 601.But wait, is there any overlap between these roots? That is, is there any ( k ) such that ( 10k = 10m + 6 ) for some integers ( k, m )? That would mean ( 10(k - m) = 6 ), which implies ( 5(k - m) = 3 ). But 5 doesn't divide 3, so there's no integer solution. Therefore, there's no overlap between the two sets of roots.So, total roots are indeed 601.Wait, but earlier I thought it was 754. Hmm, perhaps I made a mistake in the initial reasoning.Wait, let's go back. The initial reasoning was that since the function is symmetric about both 3 and 8, it's periodic with period 10. Then, from ( g(0) = 0 ), we get ( g(6) = 0 ) due to symmetry about 3, and then due to periodicity, ( g(6 + 10k) = 0 ). Similarly, reflecting over 8, we get ( g(16) = 0 ), but ( g(16) = g(6) = 0 ), which is already accounted for.So, the roots are at ( x = 10k ) and ( x = 10k + 6 ). So, in each period of 10, two roots. So, over 3000 units, 300 periods, 600 roots. But since we include both endpoints, we have an extra root at 1500, which is a multiple of 10, so total 601.But wait, in the initial problem, the interval is from -1500 to 1500, inclusive. So, the length is 3000, but the number of periods is 300, each contributing two roots, but the starting point is included, so total roots would be 300 * 2 + 1 = 601.But earlier, I thought it was 754. That must have been a mistake.Wait, let me check the initial problem again. It says "the least number of roots ( g(x) = 0 ) must have in the interval ( -1500 leq x leq 1500 )."So, perhaps the function could have more roots, but we need the minimal number. So, if the function is constructed to have exactly these roots, then 601 would be the minimal number.But wait, is 601 correct? Let me think again.Each period of 10 has two roots: one at the multiple of 10, and one at 6 more than that multiple. So, from 0 to 10, roots at 0 and 6. From 10 to 20, roots at 10 and 16, etc.Similarly, on the negative side, from -10 to 0, roots at -10 and -4. From -20 to -10, roots at -20 and -14, etc.So, in each 10-unit interval, there are two roots. So, over 3000 units, that's 300 intervals, each contributing two roots, so 600 roots. But since the interval is from -1500 to 1500, inclusive, we have to include the endpoints.At x = -1500, which is a multiple of 10, so that's a root. Similarly, x = 1500 is a multiple of 10, so that's a root. So, in total, 600 + 1 = 601 roots.Wait, but actually, when you count from -1500 to 1500, the number of intervals is 300, each contributing two roots, but the total number of roots is 300 * 2 = 600, but since we include both endpoints, we have to add one more root? Wait, no, because the interval from -1500 to 1500 is 3000 units, which is exactly 300 periods of 10. Each period contributes two roots, so 300 * 2 = 600 roots. But wait, the starting point is -1500, which is a root, and the endpoint is 1500, which is also a root. So, in the count of 600, are these endpoints included?Yes, because each period includes both endpoints. For example, the first period from -1500 to -1490 includes -1500 and -1494 as roots. The next period from -1490 to -1480 includes -1490 and -1484, and so on, until the last period from 1490 to 1500 includes 1490 and 1500 as roots. So, in total, each period contributes two roots, and there are 300 periods, so 600 roots.But wait, in this case, the interval from -1500 to 1500 includes 300 periods, each contributing two roots, so 600 roots. But in the earlier count, I considered 301 roots from multiples of 10 and 300 roots from 10k + 6, totaling 601. So, which is correct?I think the confusion arises from whether the endpoints are being double-counted or not. Let's think about it differently.The roots are at x = 10k and x = 10k + 6 for all integers k. So, in the interval from -1500 to 1500, how many such x are there?For x = 10k:The smallest k such that 10k >= -1500 is k = -150 (since 10*(-150) = -1500).The largest k such that 10k <= 1500 is k = 150 (since 10*150 = 1500).So, k ranges from -150 to 150, inclusive. That's 301 values.For x = 10k + 6:We need 10k + 6 >= -1500 => 10k >= -1506 => k >= -150.6, so k >= -150 (since k is integer).And 10k + 6 <= 1500 => 10k <= 1494 => k <= 149.4, so k <= 149.So, k ranges from -150 to 149, inclusive. That's 300 values.Therefore, total roots are 301 + 300 = 601.But wait, in the interval from -1500 to 1500, the root at x = 1500 is included, and the root at x = -1500 is included. Also, the root at x = 6 is included, and the root at x = -4 is included (since 10*(-150) + 6 = -1500 + 6 = -1494, which is within the interval).So, yes, 601 roots in total.But earlier, I thought it was 754. That must have been a mistake. So, the correct minimal number of roots is 601.Wait, but let me think again. The function is periodic with period 10, and in each period, there are two roots. So, over 300 periods, that's 600 roots. But since the interval is from -1500 to 1500, which is exactly 300 periods, the total number of roots is 600. But wait, in the count above, we have 601 roots because we include both endpoints. So, which is it?I think the confusion is whether the interval is open or closed. Since the interval is closed, including both endpoints, and both endpoints are roots, we have to include them. So, in the count of 300 periods, each contributing two roots, but the starting point is included as a root, and the ending point is included as a root. So, the total number is 300 * 2 = 600, but since the starting and ending points are both roots, we have to add 1 more root? Wait, no, because the starting point is already included in the first period, and the ending point is included in the last period.Wait, let's think of it as the number of roots in each period, including the endpoints. So, each period from a to a+10 includes two roots: a and a+6. So, the next period starts at a+10, which is already counted as a root in the previous period. So, in reality, the number of roots is 2 per period, but the starting point is shared between two periods.Wait, no, because each period is non-overlapping. Wait, no, actually, the periods are overlapping in the sense that the function is periodic, but the roots are spaced every 10 units. So, the roots are at 10k and 10k + 6, which are distinct and non-overlapping.So, in the interval from -1500 to 1500, the number of roots is 301 (from multiples of 10) + 300 (from 10k + 6) = 601.Therefore, the minimal number of roots is 601.Wait, but in the initial problem, the user wrote that the answer was 754, but that must have been incorrect.Wait, let me check the initial problem again. The user wrote:"A function ( g ) is defined for all real numbers and satisfies ( g(3 + x) = g(3 - x) ) and ( g(8 + x) = g(8 - x) ) for all ( x ). If ( g(0) = 0 ), what is the least number of roots ( g(x) = 0 ) must have in the interval ( -1500 leq x leq 1500 )."And the initial assistant answer was 754, but the user's own thought process led to 601.Wait, perhaps the initial assistant answer was incorrect. Let me think again.Given that the function is symmetric about both 3 and 8, it's periodic with period 10. Therefore, the roots are at x = 10k and x = 10k + 6 for all integers k.So, in the interval from -1500 to 1500, how many such x are there?For x = 10k:k ranges from -150 to 150, inclusive. That's 301 roots.For x = 10k + 6:k ranges from -150 to 149, inclusive. That's 300 roots.Total roots: 301 + 300 = 601.Therefore, the minimal number of roots is 601.But wait, let me think about whether the function could have more roots. The problem says "the least number of roots ( g(x) = 0 ) must have". So, we need to find the minimal number, not the maximal.So, if the function is constructed to have exactly these roots, then 601 is the minimal number. But perhaps the function could have more roots, but we need the minimal.Wait, but given the symmetries and periodicity, the function must have at least these roots. It could have more, but the minimal number is 601.Therefore, the answer should be 601.But wait, in the initial problem, the user wrote that the answer was 754, but that must have been a mistake.Wait, perhaps I made a mistake in the count. Let me recount.For x = 10k:From k = -150 to k = 150, that's 301 roots.For x = 10k + 6:From k = -150 to k = 149, that's 300 roots.Total: 301 + 300 = 601.Yes, that seems correct.Alternatively, perhaps the function has more roots due to the symmetries. For example, reflecting 0 over 3 gives 6, reflecting 6 over 8 gives 10, reflecting 10 over 3 gives 16, reflecting 16 over 8 gives 20, and so on. So, the roots are at 0, 6, 10, 16, 20, 26, etc., which are the same as 10k and 10k + 6.Therefore, the minimal number of roots is indeed 601.So, the initial answer of 754 was incorrect. The correct minimal number is 601.But wait, let me think again. The interval is from -1500 to 1500, which is 3000 units. Each period is 10 units, so 300 periods. Each period has two roots, so 600 roots. But since the interval is closed, including both endpoints, which are both roots, we have 600 + 1 = 601 roots.Yes, that makes sense.Therefore, the least number of roots is 601.

💡Okay, so I have this probability problem to solve. It says that five cards are dealt at random from a standard deck of 52 cards. I need to find the probability that the first card is an Ace, the second card is a King, the third card is a Queen, the fourth card is a Jack, and the fifth card is any card from the hearts suit. And it's without replacement, so once a card is dealt, it's not put back into the deck.Alright, let me break this down step by step. First, I know that a standard deck has 52 cards, with 13 cards in each of the four suits: hearts, diamonds, clubs, and spades. Each suit has an Ace, King, Queen, Jack, and then numbered cards from 2 to 10.So, the problem is asking for a specific sequence of cards: Ace, King, Queen, Jack, and then any heart. And each time a card is dealt, it's not replaced, so the deck size decreases by one each time.Let me think about the probability for each position one by one.1. **First card: Ace** There are 4 Aces in the deck. So, the probability that the first card is an Ace is 4 out of 52. That simplifies to 1/13. So, P(Ace first) = 4/52 = 1/13.2. **Second card: King** Now, after dealing the first card, which was an Ace, there are 51 cards left. There are still 4 Kings in the deck because the first card was an Ace, not a King. So, the probability that the second card is a King is 4 out of 51. So, P(King second) = 4/51.3. **Third card: Queen** After dealing two cards (an Ace and a King), there are 50 cards left. There are still 4 Queens in the deck because neither the Ace nor the King was a Queen. So, the probability that the third card is a Queen is 4 out of 50. So, P(Queen third) = 4/50 = 2/25.4. **Fourth card: Jack** Now, after dealing three cards (Ace, King, Queen), there are 49 cards left. There are still 4 Jacks in the deck because none of the previous three cards were Jacks. So, the probability that the fourth card is a Jack is 4 out of 49. So, P(Jack fourth) = 4/49.5. **Fifth card: Any heart** After dealing four cards (Ace, King, Queen, Jack), there are 48 cards left. Now, here's where it gets a bit tricky. The fifth card needs to be any heart. Initially, there are 13 hearts in the deck. But some of the previous four cards might have been hearts. For example, if the Ace of hearts was dealt first, then there are only 12 hearts left. Similarly, if the King of hearts, Queen of hearts, or Jack of hearts were dealt in the first four positions, that would reduce the number of hearts remaining. So, I need to consider how many hearts could have been already dealt in the first four cards. The Ace, King, Queen, and Jack could each be of any suit, including hearts. So, the maximum number of hearts that could have been removed is 4 (if all four were hearts), and the minimum is 0 (if none were hearts). Therefore, the number of hearts remaining in the deck for the fifth card could be anywhere from 13 - 4 = 9 to 13 - 0 = 13. But wait, I think I need to adjust this. Since we're dealing without replacement, and the first four cards are specific ranks (Ace, King, Queen, Jack), each of these could be of any suit. So, the number of hearts remaining depends on how many of these four ranks were hearts in the first four cards. To calculate the probability accurately, I think I need to consider all possible scenarios of how many hearts were dealt in the first four cards and then compute the expected number of hearts remaining for the fifth card. Alternatively, maybe there's a simpler way. Since each of the first four cards is equally likely to be any suit, the probability that any particular one is a heart is 1/4. So, on average, how many hearts would have been dealt in the first four cards? It would be 4 * (1/4) = 1 heart on average. But I'm not sure if this is the right approach because we're dealing with specific ranks, not random cards. Wait, maybe it's better to think about the total number of hearts remaining after four specific cards have been dealt. Since the first four cards are specific ranks (Ace, King, Queen, Jack), each of these ranks has one heart. So, there are four specific heart cards that could have been dealt in the first four positions. Therefore, the number of hearts remaining in the deck for the fifth card depends on how many of these four heart cards (Ace, King, Queen, Jack of hearts) were dealt in the first four positions. So, the number of hearts remaining is 13 minus the number of heart cards dealt in the first four positions. Therefore, the probability that the fifth card is a heart is (13 - number of heart cards dealt in first four) divided by 48. But since we don't know exactly how many heart cards were dealt in the first four positions, we need to consider all possibilities and their probabilities. This seems complicated. Maybe there's a better way. Perhaps instead of worrying about how many hearts were dealt in the first four positions, I can think about the probability that the fifth card is a heart, given that four specific cards have been removed. But I'm not sure. Maybe I should think about it as follows: The fifth card needs to be a heart, and it's being drawn from the remaining 48 cards. The number of hearts remaining is 13 minus the number of hearts already dealt in the first four cards. Since the first four cards are specific ranks (Ace, King, Queen, Jack), each of these ranks has one heart. So, the number of hearts dealt in the first four cards can be 0, 1, 2, 3, or 4. Therefore, the probability that the fifth card is a heart is the sum over k=0 to 4 of [Probability that k hearts were dealt in the first four cards] * [(13 - k)/48]. So, I need to calculate the probability of k hearts being dealt in the first four cards, for k=0,1,2,3,4. How do I calculate that? Well, the first four cards are four specific ranks: Ace, King, Queen, Jack. Each of these ranks has one heart. So, the probability that a particular one of these four cards is a heart is 1/4, since each rank has four suits. Wait, but actually, for each of these four ranks, the probability that the card dealt is a heart is 1/4, assuming that the dealing is random. So, the number of hearts dealt in the first four cards follows a binomial distribution with n=4 trials and probability p=1/4. Therefore, the probability of k hearts in the first four cards is C(4,k) * (1/4)^k * (3/4)^(4-k). So, let's calculate that for k=0 to 4. - For k=0: C(4,0)*(1/4)^0*(3/4)^4 = 1*1*(81/256) = 81/256 - For k=1: C(4,1)*(1/4)^1*(3/4)^3 = 4*(1/4)*(27/64) = 4*(27/256) = 108/256 - For k=2: C(4,2)*(1/4)^2*(3/4)^2 = 6*(1/16)*(9/16) = 6*(9/256) = 54/256 - For k=3: C(4,3)*(1/4)^3*(3/4)^1 = 4*(1/64)*(3/4) = 4*(3/256) = 12/256 - For k=4: C(4,4)*(1/4)^4*(3/4)^0 = 1*(1/256)*1 = 1/256 Now, let's verify that these probabilities sum to 1: 81 + 108 + 54 + 12 + 1 = 256, so yes, they sum to 256/256 = 1. Good, so now, for each k, the probability that k hearts were dealt in the first four cards is as above, and the number of hearts remaining is 13 - k. Therefore, the expected number of hearts remaining is the sum over k=0 to 4 of [Probability(k)] * (13 - k). But actually, we need the expected probability, which is the sum over k=0 to 4 of [Probability(k)] * [(13 - k)/48]. So, let's compute that. First, let's compute each term: - For k=0: (81/256) * (13 - 0)/48 = (81/256)*(13/48) = (81*13)/(256*48) - For k=1: (108/256) * (13 - 1)/48 = (108/256)*(12/48) = (108*12)/(256*48) - For k=2: (54/256) * (13 - 2)/48 = (54/256)*(11/48) = (54*11)/(256*48) - For k=3: (12/256) * (13 - 3)/48 = (12/256)*(10/48) = (12*10)/(256*48) - For k=4: (1/256) * (13 - 4)/48 = (1/256)*(9/48) = (1*9)/(256*48) Now, let's compute each numerator: - k=0: 81*13 = 1053 - k=1: 108*12 = 1296 - k=2: 54*11 = 594 - k=3: 12*10 = 120 - k=4: 1*9 = 9 So, the total numerator is 1053 + 1296 + 594 + 120 + 9 = Let's add them up: 1053 + 1296 = 2349 2349 + 594 = 2943 2943 + 120 = 3063 3063 + 9 = 3072 So, the total numerator is 3072, and the denominator is 256*48 = 12288. Therefore, the probability is 3072/12288. Simplifying this fraction: Divide numerator and denominator by 3072: 3072 ÷ 3072 = 1 12288 ÷ 3072 = 4 So, the probability is 1/4. Wait, that's interesting. So, the probability that the fifth card is a heart is 1/4, regardless of the first four cards. That makes sense because each card in the deck is equally likely to be in any position, and the fifth card is just another random card from the deck, so the probability it's a heart is 13/52 = 1/4. So, maybe I overcomplicated it earlier. Since the dealing is random, the fifth card is equally likely to be any of the remaining 48 cards, and the number of hearts remaining is 13 minus the number of hearts already dealt in the first four cards. But since the first four cards are specific ranks, each of which has an equal chance of being any suit, the expected number of hearts remaining is 13 - 4*(1/4) = 13 - 1 = 12. Wait, but earlier I got 1/4 as the probability, which is 12/48 = 1/4. So, that matches. Therefore, the probability that the fifth card is a heart is 1/4. So, putting it all together, the total probability is the product of the probabilities at each step: P = (4/52) * (4/51) * (4/50) * (4/49) * (1/4) Let's compute this: First, simplify each fraction: 4/52 = 1/13 4/51 remains as is. 4/50 = 2/25 4/49 remains as is. 1/4 remains as is. So, P = (1/13) * (4/51) * (2/25) * (4/49) * (1/4) Now, let's multiply these together: Multiply the numerators: 1 * 4 * 2 * 4 * 1 = 32 Multiply the denominators: 13 * 51 * 25 * 49 * 4 Let's compute the denominator step by step: 13 * 51 = 663 663 * 25 = 16575 16575 * 49 = Let's compute 16575 * 50 = 828,750, then subtract 16575 to get 828,750 - 16,575 = 812,175 812,175 * 4 = 3,248,700 So, the denominator is 3,248,700. Therefore, P = 32 / 3,248,700 Simplify this fraction: Divide numerator and denominator by 4: 32 ÷ 4 = 8 3,248,700 ÷ 4 = 812,175 So, P = 8 / 812,175 Can we simplify this further? Let's see if 8 and 812,175 have any common factors. 812,175 ÷ 8 = 101,521.875, which is not an integer, so 8 and 812,175 are coprime. Therefore, the probability is 8/812,175. But let's double-check the calculations to make sure I didn't make a mistake. Wait, when I multiplied the denominators: 13 * 51 = 663 663 * 25 = 16,575 16,575 * 49 = Let's compute 16,575 * 50 = 828,750, then subtract 16,575 to get 828,750 - 16,575 = 812,175 812,175 * 4 = 3,248,700 That seems correct. Numerators: 1 * 4 * 2 * 4 * 1 = 32 So, 32 / 3,248,700 = 8 / 812,175 Yes, that's correct. Alternatively, we can write this as a decimal to get a sense of the probability. 8 / 812,175 ≈ 0.00000985 So, approximately 0.000985%, which is very low, as expected, since it's a very specific sequence. Therefore, the final probability is 8/812,175.

💡Okay, so I have this problem here: "If the expression √(x - 4) is meaningful within the real number range, then the range of values for x is: A: x ≤ -4, B: x ≥ -4, C: x ≤ 4, D: x ≥ 4." Alright, let me try to figure this out step by step.First, I remember that when dealing with square roots, the expression inside the square root, which is called the radicand, has to be non-negative. That means it should be zero or positive because you can't take the square root of a negative number in the set of real numbers. So, for √(x - 4) to be meaningful, the expression inside the square root, which is (x - 4), must be greater than or equal to zero.So, let's write that down as an inequality: x - 4 ≥ 0. Now, I need to solve this inequality to find the range of x values that satisfy it.To solve x - 4 ≥ 0, I can add 4 to both sides of the inequality to isolate x. Adding 4 to both sides gives me x - 4 + 4 ≥ 0 + 4, which simplifies to x ≥ 4.Okay, so that means x has to be greater than or equal to 4 for the expression √(x - 4) to be meaningful in the real number system. Let me double-check that to make sure I didn't make a mistake.If x is 4, then √(4 - 4) = √0 = 0, which is fine. If x is greater than 4, say 5, then √(5 - 4) = √1 = 1, which is also fine. But if x is less than 4, say 3, then √(3 - 4) = √(-1), which is not a real number. So, that confirms that x needs to be at least 4.Looking back at the options given: A is x ≤ -4, which doesn't make sense because x needs to be greater than or equal to 4, not less than or equal to -4. B is x ≥ -4, but again, x needs to be greater than or equal to 4, not just -4. C is x ≤ 4, which is the opposite of what we need. D is x ≥ 4, which matches our solution.So, I think the correct answer is D: x ≥ 4. It's important to remember that with square roots, the radicand must be non-negative, so setting up that inequality and solving for x gives us the range of values that make the expression meaningful.I wonder if there's another way to think about this. Maybe graphing the function y = √(x - 4) could help visualize why x has to be at least 4. If I plot this function, it starts at x = 4, y = 0, and then increases as x increases. There's no part of the graph to the left of x = 4 because the square root of a negative number isn't real. So, that also shows that x must be greater than or equal to 4.Another thing to consider is if there are any exceptions or special cases. For example, what if x is exactly 4? Well, as I checked earlier, √(4 - 4) = 0, which is a real number, so x = 4 is included in the domain. What about x being a very large number, like 1000? √(1000 - 4) = √996, which is a real number, so that's fine too.I also recall that sometimes people get confused with inequalities involving square roots, especially when variables are involved. It's crucial to remember that the expression inside the square root must be non-negative, and solving that inequality gives the domain of the function.In summary, by ensuring that the radicand (x - 4) is non-negative, we set up the inequality x - 4 ≥ 0, solve for x to get x ≥ 4, and confirm that this is the correct range by checking specific values and considering the graph of the function. Therefore, the answer should be D: x ≥ 4.

💡Alright, so I've got this math problem here about something called "companion numbers." I'm not familiar with that term before, but the problem defines it for me. It says that if two numbers satisfy the equation a + b = ab, then they're called companion numbers. Okay, so for example, 3 and 1.5 are companion numbers because 3 + 1.5 equals 4.5, and 3 multiplied by 1.5 also equals 4.5. Got it.Now, the problem has two parts. Let me tackle them one by one.**Part (1): Which of the following numbers is a pair of "companion numbers" with -1?**The options are:A. -1B. 0C. 0.5D. -2So, I need to find which number, when paired with -1, satisfies the equation a + b = ab. Let's denote -1 as a, and the other number as b. So, the equation becomes:-1 + b = (-1) * bLet me solve this equation for b.Starting with:-1 + b = -bI can add b to both sides to get:-1 + b + b = 0-1 + 2b = 0Now, add 1 to both sides:2b = 1Divide both sides by 2:b = 1/2So, b equals 0.5. Looking back at the options, that's option C. So, I think the answer is C. But let me double-check by plugging it back into the original equation.Check:-1 + 0.5 = (-1) * 0.5-0.5 = -0.5Yes, that works. So, -1 and 0.5 are companion numbers.Just to be thorough, let me check the other options quickly to make sure none of them work.Option A: -1-1 + (-1) = (-1)*(-1)-2 = 1Nope, that's not true.Option B: 0-1 + 0 = (-1)*0-1 = 0Nope, not true.Option D: -2-1 + (-2) = (-1)*(-2)-3 = 2Nope, not true.So, definitely, option C is the correct answer.**Part (2): If m and n are a pair of "companion numbers," find the value of -2mn + (1/2)[3m + 2((1/2)n - m) + 3mn - 6].**Okay, this looks more complicated. Let's break it down step by step.First, since m and n are companion numbers, they satisfy the equation m + n = mn. That's given.So, we can use that fact to simplify the expression.The expression to evaluate is:-2mn + (1/2)[3m + 2((1/2)n - m) + 3mn - 6]Let me write that out again for clarity:-2mn + (1/2)[3m + 2*( (1/2)n - m ) + 3mn - 6]First, I'll simplify the innermost part: 2*( (1/2)n - m )Calculating that:2*( (1/2)n - m ) = 2*(1/2 n) - 2*m = n - 2mSo, replacing that back into the expression:-2mn + (1/2)[3m + (n - 2m) + 3mn - 6]Now, let's simplify inside the brackets:3m + n - 2m + 3mn - 6Combine like terms:(3m - 2m) + n + 3mn - 6 = m + n + 3mn - 6So, the expression becomes:-2mn + (1/2)(m + n + 3mn - 6)Now, let's distribute the (1/2) across the terms inside the brackets:-2mn + (1/2)*m + (1/2)*n + (1/2)*3mn - (1/2)*6Calculating each term:-2mn + (m/2) + (n/2) + (3mn/2) - 3Now, let's combine like terms. First, let's handle the mn terms:-2mn + (3mn/2) = (-4mn/2 + 3mn/2) = (-mn/2)Then, the m and n terms:(m/2) + (n/2) = (m + n)/2And the constant term:-3So, putting it all together:(-mn/2) + (m + n)/2 - 3Now, remember that m and n are companion numbers, so m + n = mn. Let's substitute mn for m + n wherever possible.First, let's look at (m + n)/2. Since m + n = mn, this becomes mn/2.So, replacing (m + n)/2 with mn/2:(-mn/2) + (mn/2) - 3Now, let's combine the mn terms:(-mn/2 + mn/2) = 0So, we're left with:0 - 3 = -3Therefore, the value of the expression is -3.Wait, let me double-check that to make sure I didn't make any mistakes.Starting from the simplified expression:(-mn/2) + (m + n)/2 - 3Since m + n = mn, substitute:(-mn/2) + (mn)/2 - 3Which simplifies to:0 - 3 = -3Yes, that seems correct. So, the final answer is -3.Just to recap, the key steps were:1. Recognize that m + n = mn.2. Substitute mn for m + n in the expression.3. Simplify the expression step by step, carefully combining like terms.4. Realize that the mn terms cancel out, leaving just the constant term.So, I'm confident that the answer is -3.**Final Answer**(1) boxed{C}(2) boxed{-3}

💡Okay, so I have this problem where I need to figure out how many distinct ways I can arrange my six keys on a keychain. The conditions are that my house key needs to be next to my car key, and my office key needs to be next to my bike key. Also, two arrangements are considered the same if you can rotate or flip the keychain to make them look identical. Hmm, that sounds a bit tricky, but let me try to break it down step by step.First, I know that arranging keys on a keychain is different from arranging them in a straight line because a keychain is circular. In a straight line, the number of arrangements is just factorial of the number of keys, but on a circle, it's different because rotations and reflections can make some arrangements identical. I remember that for circular arrangements, the number of distinct arrangements is (n-1)! because fixing one position accounts for rotations, and then dividing by 2 accounts for reflections. So, for six keys, it would be (6-1)! / 2 = 120 / 2 = 60. But that's without any restrictions.But in this problem, I have specific restrictions: house key next to car key and office key next to bike key. So, I need to adjust my calculation to account for these constraints.Let me think about how to handle these restrictions. When two keys must be next to each other, it's often helpful to treat them as a single unit or "block." So, if I consider the house key and car key as one block, and the office key and bike key as another block, then I effectively reduce the number of items I'm arranging. Instead of six individual keys, I now have four items: the HC block, the OB block, and the remaining two keys, let's call them Key1 and Key2.Now, I need to arrange these four items on a keychain. As I thought earlier, the number of distinct arrangements for four items on a keychain is (4-1)! / 2 = 6 / 2 = 3. Wait, that doesn't seem right. Let me double-check. For n items on a keychain, the formula is (n-1)! / 2. So, for n=4, it's (4-1)! / 2 = 6 / 2 = 3. Hmm, that seems too low. Maybe I'm missing something.Wait, no, actually, the formula is correct. For four distinct items on a keychain, there are 3 distinct arrangements. But in this case, the HC and OB blocks are not just single items; they can be arranged in two different ways each. For example, the HC block can be HC or CH, and the OB block can be OB or BO. So, each block has 2 possible internal arrangements.Therefore, for each of the 3 distinct arrangements of the four items (HC, OB, Key1, Key2), there are 2 possibilities for HC and 2 possibilities for OB. So, the total number of arrangements should be 3 * 2 * 2 = 12.But wait, that doesn't seem right either because I remember that when treating blocks as units, the formula might be different. Let me think again. If I have four units, the number of circular arrangements is (4-1)! = 6, and since reflections are considered the same, we divide by 2, giving 3. Then, for each of these 3 arrangements, we have 2 possibilities for HC and 2 for OB, so 3 * 2 * 2 = 12. That seems correct.But hold on, I think I might have made a mistake in considering the blocks. Let me try a different approach. Instead of treating the blocks as single units, maybe I should consider the problem as arranging the keys with specific adjacency constraints.So, we have six keys: H, C, O, B, Key1, Key2. We need H next to C and O next to B. Let's first consider the HC pair and the OB pair. Each pair can be arranged in 2 ways: HC or CH, and OB or BO.Now, if I fix the position of one pair to account for the circular nature, say fix HC in a certain position, then I need to arrange the remaining keys: OB, Key1, Key2. But OB is another pair, so it's like arranging three items: OB, Key1, Key2. However, since it's a keychain, fixing one position, the number of arrangements is (3-1)! = 2, and considering reflections, it's 2 / 2 = 1. Wait, that doesn't make sense because we have two possible internal arrangements for OB.Wait, maybe I'm overcomplicating it. Let's try to visualize it. If I fix the HC pair in a position, then the remaining four positions need to be filled with OB, Key1, Key2, and the other end of the HC pair. But since it's a keychain, the arrangement is circular, so fixing HC doesn't account for all symmetries.Alternatively, maybe I should use the concept of treating the pairs as single entities and then calculate the number of circular arrangements for these entities. So, we have four entities: HC, OB, Key1, Key2. The number of circular arrangements for four entities is (4-1)! = 6. Since reflections are considered the same, we divide by 2, giving 3. Then, for each of these 3 arrangements, we have 2 possibilities for HC and 2 for OB, so 3 * 2 * 2 = 12.But earlier, I thought the answer was 24. Hmm, where is the discrepancy? Maybe I'm not accounting for something. Let me think again.Wait, perhaps I should not fix any position when calculating the circular arrangements. The formula for circular arrangements is (n-1)! / 2 for indistinct arrangements considering rotations and reflections. So, for four entities, it's (4-1)! / 2 = 6 / 2 = 3. Then, multiplying by the internal arrangements of the pairs: 2 * 2 = 4. So, total arrangements: 3 * 4 = 12.But I'm confused because I initially thought it was 24. Maybe I need to consider that the keychain can be flipped, so each arrangement is counted twice. Wait, no, the formula already accounts for reflections by dividing by 2. So, 3 * 4 = 12 should be correct.But let me try another approach to verify. Suppose I don't treat the pairs as single units. Instead, I calculate the total number of arrangements without restrictions and then divide by the symmetries. But that might be more complicated.Alternatively, I can think of the problem as arranging the keys with the given adjacency constraints. So, first, fix the position of one key to eliminate rotational symmetry. Let's fix the house key (H) at a position. Then, the car key (C) must be either to the left or right of H. So, two choices for C.Now, we have four remaining keys: O, B, Key1, Key2. But O and B must be adjacent. So, let's treat O and B as a pair. Now, we have three items to arrange: OB, Key1, Key2. Since we've fixed H and C, the keychain is now effectively a straight line from H to C, with the remaining keys arranged in the remaining four positions. But since it's a keychain, the arrangement is circular, so fixing H and C doesn't account for all symmetries.Wait, maybe I should fix H and C in a specific position to eliminate rotational symmetry. So, fix H at position 1 and C at position 2. Now, we have four remaining positions: 3, 4, 5, 6. We need to arrange O, B, Key1, Key2 such that O and B are adjacent.So, how many ways can we arrange O and B as a pair in the remaining four positions? Let's consider the pair OB or BO. The pair can be placed in positions 3-4, 4-5, or 5-6. So, three possible positions for the OB pair. For each position, the pair can be OB or BO, so 2 possibilities. Then, the remaining two keys, Key1 and Key2, can be arranged in the remaining two positions in 2! = 2 ways.So, total arrangements: 3 (positions) * 2 (internal arrangements of OB) * 2 (arrangements of Key1 and Key2) = 12.But wait, we fixed H at position 1 and C at position 2, so we've accounted for rotational symmetry. However, we haven't accounted for reflection symmetry. Since the keychain can be flipped, arrangements that are mirror images are considered the same. So, do we need to divide by 2?Wait, in this case, since we've fixed H and C in specific positions, flipping the keychain would move H to position 6 and C to position 5, which is a different arrangement because we've fixed H at position 1. So, perhaps we don't need to divide by 2 here because we've already fixed the positions to account for rotational symmetry, and reflections would result in different fixed positions, which we haven't considered.Hmm, this is getting confusing. Maybe I should consider that by fixing H and C, we've eliminated rotational symmetry, but reflection symmetry can still create equivalent arrangements. So, perhaps some of the 12 arrangements are equivalent under reflection.Wait, but since we've fixed H and C, reflecting the keychain would move H to position 6 and C to position 5, which is a different arrangement because we've fixed H at position 1. Therefore, reflections would result in different fixed positions, which we haven't considered. So, in this case, we don't need to divide by 2 because we've already fixed the positions to account for rotational symmetry, and reflections would result in different fixed positions, which are not equivalent.Therefore, the total number of distinct arrangements is 12.But earlier, I thought it was 24. Hmm, maybe I made a mistake in the first approach. Let me try to reconcile the two methods.In the first approach, treating HC and OB as blocks, we had four entities: HC, OB, Key1, Key2. The number of circular arrangements is (4-1)! / 2 = 3. Then, multiplying by the internal arrangements of the blocks: 2 * 2 = 4. So, total arrangements: 3 * 4 = 12.In the second approach, fixing H and C, we had 12 arrangements without considering reflection. But since reflections would result in different fixed positions, we don't need to divide by 2, so total arrangements remain 12.Wait, but in the first approach, we considered circular arrangements of four entities, which already accounts for rotations and reflections. So, why in the second approach, we didn't need to divide by 2? Because in the second approach, we fixed H and C, which eliminated rotational symmetry, but reflections would still create equivalent arrangements. However, since we fixed H and C, reflections would move H and C to different positions, which we haven't considered, so perhaps we don't need to divide by 2.But I'm getting conflicting results. Maybe I need to think differently.Let me try to calculate the total number of arrangements without considering the keychain's circular nature first, and then adjust for it.So, if I have six keys, with H next to C and O next to B, how many linear arrangements are there?First, treat HC as a block and OB as a block. So, we have four blocks: HC, OB, Key1, Key2. The number of ways to arrange these four blocks in a line is 4! = 24. Each block can be arranged internally in 2 ways, so total arrangements: 24 * 2 * 2 = 96.But since it's a keychain, which is circular, we need to adjust for circular permutations. The number of circular permutations for n items is (n-1)! So, for four blocks, it would be (4-1)! = 6. Then, considering reflections, we divide by 2, giving 3. Then, multiply by the internal arrangements: 2 * 2 = 4. So, total arrangements: 3 * 4 = 12.Wait, that's consistent with the first approach. So, the total number of distinct arrangements is 12.But earlier, when I fixed H and C, I also got 12. So, maybe 12 is the correct answer.But I'm still confused because I thought the answer was 24. Maybe I need to consider that when treating the blocks as single units, the formula is different.Wait, let me think again. If I have four blocks, the number of circular arrangements is (4-1)! = 6. Since reflections are considered the same, we divide by 2, giving 3. Then, for each of these 3 arrangements, we have 2 possibilities for HC and 2 for OB, so 3 * 2 * 2 = 12.Alternatively, if I don't consider the keychain's circular nature, the number of linear arrangements is 4! * 2 * 2 = 96. To convert to circular arrangements, we divide by n, which is 4, giving 24. But since reflections are considered the same, we divide by 2, giving 12.Yes, that makes sense. So, the total number of distinct arrangements is 12.Wait, but in the initial problem, the user mentioned that two arrangements are not considered different if they can be made the same by reflection or rotation. So, the formula is (n-1)! / 2 for circular arrangements considering reflections.So, for four blocks, it's (4-1)! / 2 = 3. Then, multiply by the internal arrangements: 2 * 2 = 4. So, total arrangements: 3 * 4 = 12.Therefore, the answer should be 12.But wait, in the initial thought process, I thought it was 24. Maybe I made a mistake there. Let me check.In the initial thought process, I considered treating HC and OB as blocks, reducing the problem to four items. Then, the number of circular arrangements is (4-1)! = 6. Then, considering reflections, we divide by 2, giving 3. Then, multiply by the internal arrangements: 2 * 2 = 4. So, total arrangements: 3 * 4 = 12.Yes, that's consistent. So, the correct answer is 12.But wait, in the initial problem, the user mentioned that the answer was 24, but in the thought process, I arrived at 12. So, maybe the user made a mistake in their initial thought process.Alternatively, perhaps I'm missing something. Let me try to think differently.Suppose I don't treat the pairs as blocks. Instead, I calculate the total number of arrangements where H is next to C and O is next to B.In a circular arrangement, the number of ways to arrange n items is (n-1)! / 2. For six keys, it's (6-1)! / 2 = 60.Now, the number of arrangements where H is next to C: treat HC as a block, so we have five items: HC, O, B, Key1, Key2. The number of circular arrangements is (5-1)! / 2 = 24 / 2 = 12. Then, multiply by 2 for the internal arrangement of HC, giving 24.Similarly, the number of arrangements where O is next to B: treat OB as a block, so we have five items: H, C, OB, Key1, Key2. The number of circular arrangements is (5-1)! / 2 = 24 / 2 = 12. Then, multiply by 2 for the internal arrangement of OB, giving 24.But we need both conditions: H next to C and O next to B. So, we need to find the intersection of these two sets.Using the principle of inclusion-exclusion, the total number of arrangements where H is next to C or O is next to B is 24 + 24 - arrangements where both H is next to C and O is next to B.But we need the number of arrangements where both H is next to C and O is next to B. So, let's calculate that.Treat HC as a block and OB as a block. Now, we have four blocks: HC, OB, Key1, Key2. The number of circular arrangements is (4-1)! / 2 = 3. Then, multiply by 2 for HC and 2 for OB, giving 3 * 2 * 2 = 12.Therefore, the number of arrangements where both H is next to C and O is next to B is 12.So, the total number of distinct arrangements is 12.Therefore, the correct answer is 12.But wait, in the initial problem, the user mentioned that the answer was 24. So, maybe I'm missing something. Let me check again.Alternatively, perhaps the user considered that the keychain can be flipped, so each arrangement is counted twice, but in the formula, we already divided by 2 to account for reflections. So, maybe the correct answer is 12.Alternatively, maybe the user considered that the keychain can be flipped, but in the formula, we already accounted for that by dividing by 2. So, the correct answer is 12.Wait, but in the initial thought process, the user arrived at 24, but in my calculation, I arrived at 12. So, perhaps the user made a mistake.Alternatively, maybe I made a mistake in considering the number of circular arrangements. Let me think again.For four blocks, the number of circular arrangements is (4-1)! = 6. Since reflections are considered the same, we divide by 2, giving 3. Then, multiply by 2 for HC and 2 for OB, giving 12.Yes, that seems correct.Alternatively, if I don't consider reflections, the number of circular arrangements is 6, and then multiply by 4, giving 24. But since reflections are considered the same, we divide by 2, giving 12.Therefore, the correct answer is 12.But wait, in the initial problem, the user mentioned that the answer was 24. So, maybe the user didn't account for reflections correctly.Alternatively, perhaps the user considered that the keychain can be flipped, so each arrangement is counted twice, but in the formula, we already divided by 2 to account for reflections. So, the correct answer is 12.Therefore, I think the correct answer is 12.