💡Okay, so I have this problem where Alice has 24 apples, and she wants to share them with Becky and Chris. The condition is that each of the three people—Alice, Becky, and Chris—must have at least two apples. I need to figure out how many ways this can be done. The answer choices are given as (A) 105, (B) 114, (C) 190, (D) 210, and (E) 380.Alright, let me break this down. First, it's a problem about distributing apples among three people, each getting at least two apples. So, this is a combinatorics problem, specifically about partitions or combinations with constraints.I remember that when distributing identical items (in this case, apples) to distinct recipients (Alice, Becky, Chris), we can use the stars and bars method. But since each person must have at least two apples, we can't have any of them getting zero or one apple. So, we need to adjust the problem to account for this minimum.Let me think. If each person must have at least two apples, maybe I can subtract the minimum number of apples each person gets from the total and then distribute the remaining apples without any restrictions. That way, each person will automatically have at least two apples.So, if each of the three people gets two apples, that's a total of 2 * 3 = 6 apples. Since Alice has 24 apples, subtracting these 6 apples leaves us with 24 - 6 = 18 apples to distribute freely among the three people.Now, the problem reduces to finding the number of ways to distribute 18 identical apples among 3 distinct people, where each person can receive any number of apples, including zero. This is a classic stars and bars problem.The formula for the number of ways to distribute n identical items into k distinct groups is given by the combination formula C(n + k - 1, k - 1). In this case, n is 18 apples, and k is 3 people. So, plugging into the formula, we get C(18 + 3 - 1, 3 - 1) = C(20, 2).Calculating C(20, 2), which is the number of combinations of 20 things taken 2 at a time. The formula for combinations is C(n, r) = n! / (r! * (n - r)!). So, C(20, 2) = 20! / (2! * 18!) = (20 * 19) / (2 * 1) = 380 / 2 = 190.Wait, that gives me 190, which is option (C). Hmm, but let me double-check my reasoning because sometimes it's easy to make a mistake in combinatorics.So, starting again: Each person must have at least two apples. So, we subtract 2 apples for each person, which is 6 apples total. That leaves 18 apples to distribute without restrictions. Using stars and bars, the number of ways is C(18 + 3 - 1, 3 - 1) = C(20, 2). Calculating that gives 190, which is one of the answer choices, specifically (C).Alternatively, I can think of it as arranging 18 stars (apples) and 2 bars (dividers between Alice, Becky, and Chris). The number of ways to arrange these is the same as choosing 2 positions out of 20 (18 stars + 2 bars), which is C(20, 2) = 190.Yes, that seems correct. So, the answer should be 190, which is option (C).**Final Answer**boxed{C}

💡First, determine the total number of possible choices for both students A and B. Since each student can choose from three interest groups, the total number of combinations is (3 times 3 = 9).Next, identify the number of favorable outcomes where A and B are not in the same interest group. For each choice that A makes, B has two other interest groups to choose from. Therefore, the number of favorable outcomes is (3 times 2 = 6).Finally, calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes: (frac{6}{9} = frac{2}{3}).

💡Alright, so I have this problem here about linear regression, and I need to figure out which equation is the correct one given some information. Let me try to break it down step by step.First, the problem says that variables ( x ) and ( y ) are linearly negatively correlated. Hmm, okay. I remember that negative correlation means that as one variable increases, the other decreases. So, if ( x ) goes up, ( y ) should go down, and vice versa. That makes sense.Next, it gives me the sample means: ( overline{x} = 3 ) and ( overline{y} = 3.5 ). I recall that in linear regression, the regression line always passes through the point ( (overline{x}, overline{y}) ). That's a key point. So whatever regression equation we come up with, when we plug in ( x = 3 ), the result should be ( y = 3.5 ). That seems important for checking the options.Now, looking at the options:A: ( y = 0.4x + 2.4 )B: ( y = 2x + 2.4 )C: ( y = -2x + 9.5 )D: ( y = -0.3x + 4.4 )Since ( x ) and ( y ) are negatively correlated, the slope of the regression line should be negative. That means options A and B can be eliminated right away because their slopes are positive (0.4 and 2, respectively). So, we're left with options C and D.Now, let's check if these lines pass through the point ( (3, 3.5) ). For option C: ( y = -2x + 9.5 ). Let me plug in ( x = 3 ):( y = -2(3) + 9.5 = -6 + 9.5 = 3.5 ). Perfect, that's exactly the mean we were given. So, option C satisfies that condition.Now, let's check option D: ( y = -0.3x + 4.4 ). Plugging in ( x = 3 ):( y = -0.3(3) + 4.4 = -0.9 + 4.4 = 3.5 ). Wait, that also gives ( y = 3.5 ). Hmm, so both C and D pass through the mean point. But we only have one correct answer, so there must be another way to differentiate between them.I think the key here is the strength of the correlation. The problem just says they're negatively correlated, but it doesn't specify the strength. However, in regression, the slope also relates to the correlation coefficient. A steeper slope (more negative) would indicate a stronger negative correlation, while a shallower slope (closer to zero) would indicate a weaker negative correlation.Looking at the options, C has a slope of -2, which is quite steep, while D has a slope of -0.3, which is much shallower. Without more information about the correlation coefficient, I can't be sure which slope is correct. But maybe there's another consideration.Wait, the problem says "could be" the regression equation. So, it's asking which one is possible, not necessarily the only one. Since both C and D pass through the mean point and have negative slopes, both could technically be regression equations for negatively correlated data. But the options only give one correct answer, so perhaps I'm missing something.Let me think again. Maybe the question is testing whether the regression line passes through the mean point, which both do, but also whether the slope makes sense given the means. Let me calculate the intercepts to see if they make sense.For option C: ( y = -2x + 9.5 ). The y-intercept is 9.5, which is quite high. If ( x = 0 ), ( y ) would be 9.5. That seems possible, but let's see how it relates to the mean.For option D: ( y = -0.3x + 4.4 ). The y-intercept is 4.4, which is lower than 9.5. If ( x = 0 ), ( y ) would be 4.4. That also seems possible.I'm not sure if the y-intercept is a factor here. Maybe I should consider the coefficient of determination or something else, but I don't have that information. The problem doesn't provide data points or variances, so I can't calculate the exact slope.Wait, maybe I should consider the fact that the regression line is the best fit, minimizing the sum of squared errors. But without data points, I can't compute that either. So, perhaps the question is only testing the two conditions: negative slope and passing through the mean point.Since both C and D satisfy those, but the options only have one correct answer, maybe I made a mistake earlier. Let me double-check my calculations.For option C: ( y = -2(3) + 9.5 = -6 + 9.5 = 3.5 ). Correct.For option D: ( y = -0.3(3) + 4.4 = -0.9 + 4.4 = 3.5 ). Correct.Hmm, so both are correct in terms of passing through the mean. Maybe the question expects us to choose the one with the steeper slope because it's more likely to be the regression line for negatively correlated data, but that's an assumption.Alternatively, perhaps the question is testing the understanding that the regression line must pass through the mean, and the negative slope is necessary. So, both C and D are possible, but since only one can be chosen, maybe the steeper slope is more indicative of a negative correlation.But I'm not entirely sure. Maybe I should look back at the problem statement. It says "could be" the regression equation, implying that only one of them is possible. Perhaps there's another condition I'm missing.Wait, the problem mentions that ( x ) and ( y ) are linearly negatively correlated, but it doesn't specify anything else. So, both C and D are valid regression lines for negatively correlated data passing through the mean. But since the options only have one correct answer, maybe the intended answer is C because it has a more negative slope, making it a stronger negative correlation, which is more typical in such problems.Alternatively, maybe the question expects us to recognize that the slope is determined by the covariance and variances, but without that data, we can't compute it exactly. So, perhaps the question is only testing the two conditions: negative slope and passing through the mean, which both C and D satisfy.But since the options only have one correct answer, I think the intended answer is C because it has a steeper slope, which is more commonly associated with a negative correlation in such multiple-choice questions.Wait, but I'm not entirely confident. Maybe I should consider that the slope is related to the ratio of the standard deviations. If the standard deviation of ( y ) is larger than that of ( x ), the slope could be steeper. But without that information, it's hard to say.Alternatively, maybe the question is testing the understanding that the regression line must pass through the mean, and the negative slope is necessary, so both C and D are correct, but since only one can be chosen, perhaps the steeper slope is the answer.I think I've overcomplicated this. The key points are: negative slope and passing through the mean. Both C and D satisfy that, but since the options only have one correct answer, I think the intended answer is C because it has a more negative slope, making it a stronger negative correlation, which is more typical in such problems.Wait, but I'm still not sure. Maybe I should look for another way to differentiate between C and D. Let me think about the y-intercepts. If ( x = 0 ), for C, ( y = 9.5 ), and for D, ( y = 4.4 ). Without knowing the range of ( x ), it's hard to say which is more reasonable. But if ( x ) is around 3, then ( y ) being 9.5 when ( x = 0 ) might be more of an extrapolation, while 4.4 is closer to the mean.Hmm, I'm not sure. Maybe the question expects us to recognize that the slope is determined by the covariance and variances, but without that data, we can't compute it exactly. So, perhaps the question is only testing the two conditions: negative slope and passing through the mean, which both C and D satisfy.But since the options only have one correct answer, I think the intended answer is C because it has a steeper slope, which is more commonly associated with a negative correlation in such multiple-choice questions.Wait, but I'm still not confident. Maybe I should consider that the slope is related to the ratio of the standard deviations. If the standard deviation of ( y ) is larger than that of ( x ), the slope could be steeper. But without that information, it's hard to say.Alternatively, maybe the question is testing the understanding that the regression line must pass through the mean, and the negative slope is necessary, so both C and D are correct, but since only one can be chosen, perhaps the steeper slope is the answer.I think I've thought about this enough. I'll go with option C because it has a negative slope and passes through the mean, and it's more likely to be the intended answer.

💡Okay, so I have this function f(x) = 3 ln x - x² + (a - 1/2)x, and I need to find the range of real numbers a such that f(x) has a maximum value on the interval (1, 3). Hmm, okay. Let me think about how to approach this.First, I remember that to find maxima or minima of a function on an interval, we usually look at critical points where the derivative is zero or undefined, and also check the endpoints. But since the interval is open, (1, 3), we don't include the endpoints. So, the maximum must occur at a critical point inside the interval.So, I should start by finding the derivative of f(x). Let me compute f'(x):f'(x) = d/dx [3 ln x - x² + (a - 1/2)x]The derivative of 3 ln x is 3/x. The derivative of -x² is -2x. The derivative of (a - 1/2)x is just (a - 1/2). So putting it all together:f'(x) = 3/x - 2x + (a - 1/2)Okay, so f'(x) = 3/x - 2x + (a - 1/2). Now, to find critical points, I need to set f'(x) = 0 and solve for x:3/x - 2x + (a - 1/2) = 0Hmm, that's a bit tricky. Let me rearrange the equation:3/x - 2x = -(a - 1/2)Or,3/x - 2x = (1/2 - a)I can write this as:(3 - 2x²)/x = (1/2 - a)So,(3 - 2x²)/x = (1/2 - a)Let me solve for a:a = 1/2 - (3 - 2x²)/xSimplify the right-hand side:a = 1/2 - 3/x + 2xSo,a = 2x - 3/x + 1/2Hmm, so for each x in (1, 3), there's a corresponding a that makes f'(x) = 0. But I need to find the range of a such that there's a maximum in (1, 3). So, perhaps I need to find the range of a for which the equation f'(x) = 0 has a solution in (1, 3), and that solution is a maximum.Wait, but how do I ensure that the critical point is a maximum? I think I need to check the second derivative or use the first derivative test.Let me compute the second derivative f''(x):f''(x) = d/dx [3/x - 2x + (a - 1/2)]The derivative of 3/x is -3/x². The derivative of -2x is -2. The derivative of a constant is zero. So,f''(x) = -3/x² - 2Since x is in (1, 3), x² is positive, so -3/x² is negative, and -2 is also negative. Therefore, f''(x) is always negative on (1, 3). That means any critical point in (1, 3) is a local maximum.So, as long as f'(x) = 0 has a solution in (1, 3), that solution will be a local maximum. Therefore, I just need to find the range of a such that f'(x) = 0 has at least one solution in (1, 3).So, going back to the equation:a = 2x - 3/x + 1/2Let me define a function g(x) = 2x - 3/x + 1/2. Then, a = g(x). So, I need to find the range of g(x) as x varies over (1, 3). The range of g(x) will give me the possible values of a for which f'(x) = 0 has a solution in (1, 3).Therefore, I need to find the minimum and maximum values of g(x) on (1, 3).To find the range of g(x), I can analyze its behavior on the interval (1, 3). Let's compute g(x) at the endpoints and see if it has any extrema inside the interval.First, compute the limit as x approaches 1 from the right:lim(x→1+) g(x) = 2(1) - 3/1 + 1/2 = 2 - 3 + 0.5 = -0.5Similarly, compute the limit as x approaches 3 from the left:lim(x→3-) g(x) = 2(3) - 3/3 + 1/2 = 6 - 1 + 0.5 = 5.5So, as x approaches 1 from the right, g(x) approaches -0.5, and as x approaches 3 from the left, g(x) approaches 5.5.Now, let's check if g(x) has any critical points in (1, 3). To do that, compute the derivative of g(x):g'(x) = d/dx [2x - 3/x + 1/2] = 2 + 3/x²Since 2 + 3/x² is always positive for all x ≠ 0, g(x) is strictly increasing on (1, 3). Therefore, g(x) has no critical points in (1, 3); it's always increasing.Therefore, the range of g(x) on (1, 3) is (-0.5, 5.5). So, a must be in (-0.5, 5.5) for f'(x) = 0 to have a solution in (1, 3), which in turn means f(x) has a maximum in (1, 3).Wait, but let me double-check. Since g(x) is strictly increasing, its minimum value is approached as x approaches 1 from the right, which is -0.5, and its maximum value is approached as x approaches 3 from the left, which is 5.5. So, a must be greater than -0.5 and less than 5.5.Therefore, the range of a is (-1/2, 11/2), since 5.5 is equal to 11/2.Let me just verify this conclusion. If a is greater than -1/2 and less than 11/2, then the equation f'(x) = 0 has a solution in (1, 3), and since f''(x) is negative there, it's a maximum. So, yes, that makes sense.I think that's the correct range for a.**Final Answer**The range of real number ( a ) is boxed{left( -frac{1}{2}, frac{11}{2} right)}.

💡First, I need to simplify the expression ( 2(x - 2y)^2 - (2y + x)(-2y + x) ). I'll start by expanding each part separately. For the first term, ( 2(x - 2y)^2 ), I'll expand the square:[(x - 2y)^2 = x^2 - 4xy + 4y^2]Multiplying by 2 gives:[2x^2 - 8xy + 8y^2]Next, for the second term, ( (2y + x)(-2y + x) ), I'll use the distributive property:[(2y + x)(-2y + x) = -4y^2 + 2xy - 2xy + x^2 = x^2 - 4y^2]Now, I'll combine the simplified terms:[2x^2 - 8xy + 8y^2 - (x^2 - 4y^2) = 2x^2 - 8xy + 8y^2 - x^2 + 4y^2 = x^2 - 8xy + 12y^2]With the expression simplified, I'll substitute ( x = 1 ) and ( y = 1 ):[x^2 - 8xy + 12y^2 = 1^2 - 8(1)(1) + 12(1)^2 = 1 - 8 + 12 = 5]Therefore, the final answer is 5.

💡Alright, so I've got this problem here about 1001 people sitting around a round table. Each person is either a knight, who always tells the truth, or a liar, who always lies. The condition given is that next to each knight, there's exactly one liar, and next to each liar, there's exactly one knight. I need to find the minimum number of knights that can be sitting at the table.Okay, let's break this down. First, knights and liars alternate around the table because each knight has a liar next to them and each liar has a knight next to them. So, the seating must follow a pattern like K-L-K-L-K-L... or L-K-L-K-L-K... around the table.Since the table is round, the pattern has to loop back on itself. That means the number of knights and liars must be such that the pattern doesn't break when it comes back to the starting point. If there are an equal number of knights and liars, that would work perfectly because the pattern would alternate smoothly without any conflicts.But wait, there are 1001 people in total, which is an odd number. If we try to have an equal number of knights and liars, we'd end up with one extra person. That extra person would disrupt the alternating pattern because they'd have to sit next to someone of the same type, which violates the given condition.So, to maintain the alternating pattern with an odd number of people, one type must have one more person than the other. Since we're looking for the minimum number of knights, we want as few knights as possible. That means we should have one more liar than knights.Let's denote the number of knights as K and the number of liars as L. We know that:K + L = 1001And since we need one more liar than knights:L = K + 1Substituting the second equation into the first:K + (K + 1) = 1001Simplifying:2K + 1 = 1001Subtract 1 from both sides:2K = 1000Divide both sides by 2:K = 500So, there are 500 knights and 501 liars. But wait, the problem asks for the minimum number of knights. Is 500 the minimum?Hold on, if we have 500 knights, each knight has exactly one liar next to them, and each liar has exactly one knight next to them. That seems to satisfy the conditions. But is there a way to have even fewer knights?Let me think. If we try to have fewer knights, say 499, then the number of liars would be 502. But then, each knight still needs to have exactly one liar next to them, and each liar needs to have exactly one knight next to them. With 499 knights and 502 liars, we can't maintain the alternating pattern because the number of liars exceeds the number of knights by more than one. This would cause some liars to be next to other liars, which violates the condition.Therefore, 500 knights seem to be the minimum number that allows the alternating pattern to hold true without any conflicts. Each knight has one liar next to them, and each liar has one knight next to them, and the total adds up to 1001 people.Wait, but earlier I thought that having one more liar than knights would mean 501 liars and 500 knights. But the problem asks for the minimum number of knights, so 500 is the answer. However, I just realized that if we start with a liar, the pattern would be L-K-L-K... and end with a liar, which would require the number of liars to be one more than knights. So, 501 liars and 500 knights.But does that mean the minimum number of knights is 500? Yes, because if we try to have fewer knights, the pattern breaks. So, 500 is indeed the minimum number of knights needed to satisfy the given conditions.Wait again, I'm getting confused. Let me double-check. If we have 500 knights, each knight needs a liar next to them, so we need at least 500 liars. But since the table is round and the pattern alternates, the number of liars must be equal to the number of knights or one more. Since 1001 is odd, it must be one more. So, liars are 501, knights are 500.But the problem asks for the minimum number of knights. So, 500 is the minimum. Wait, but earlier I thought 501 was the answer. I need to clarify this.Let's consider the pattern. If we start with a knight, the sequence would be K-L-K-L... and since 1001 is odd, it would end with a knight, making the number of knights one more than liars. But we want the minimum number of knights, so we should start with a liar. Then the sequence would be L-K-L-K... and end with a liar, making the number of liars one more than knights.Therefore, the number of liars is 501, and the number of knights is 500. So, the minimum number of knights is 500.But wait, the problem says "next to each knight there is exactly one liar, and next to each liar there is exactly one knight." If we have 500 knights and 501 liars, each knight has exactly one liar next to them, and each liar has exactly one knight next to them. That works.But if we try to have fewer knights, say 499, then we would have 502 liars. Each knight would still need one liar next to them, but the liars would have to be arranged such that each liar has exactly one knight next to them. However, with 502 liars and 499 knights, it's impossible to arrange them in an alternating pattern because the number of liars exceeds the number of knights by more than one. This would force some liars to be next to other liars, which violates the condition.Therefore, 500 knights is indeed the minimum number required to satisfy the given conditions.