💡First, I need to determine the total weight increase for the group of 12 people when the average weight increases by 2.7 kg. This can be calculated by multiplying the number of people by the increase in average weight.Next, I'll find the total weight of the three individuals who were replaced by adding their individual weights together.Finally, to find the total weight of the three new persons, I'll add the total weight increase to the total weight of the three individuals who were replaced. This will give me the combined weight of the new individuals.

💡To determine the number of distinct three-digit even numbers that can be formed using the digits 0, 1, 2, 3, and 4 without repetition, we can break down the problem into two main scenarios:1. **Units Digit is 0:** - If the units digit is fixed as 0, we need to choose the hundreds and tens digits from the remaining digits (1, 2, 3, 4). - The hundreds digit can be any of the 4 remaining digits. - The tens digit can be any of the 3 remaining digits after choosing the hundreds digit. - This gives us (4 times 3 = 12) possible numbers.2. **Units Digit is 2 or 4:** - If the units digit is either 2 or 4, we have 2 choices for the units digit. - The hundreds digit cannot be 0, so we choose it from the remaining 3 non-zero digits. - The tens digit can be any of the remaining 2 digits after choosing the hundreds digit. - This gives us (2 times 3 times 2 = 12) possible numbers.Adding the results from both scenarios, we get a total of (12 + 12 = 24) distinct three-digit even numbers.

💡Okay, so I have this expression to evaluate: (frac{tan frac{pi}{4} + i}{tan frac{pi}{4} - i}). Hmm, I remember that (tan frac{pi}{4}) is 1 because the tangent of 45 degrees is 1. So, substituting that in, the expression becomes (frac{1 + i}{1 - i}). Alright, now I need to simplify this complex fraction. I think the way to do this is by multiplying the numerator and the denominator by the conjugate of the denominator. The conjugate of (1 - i) is (1 + i). So, I'll multiply both the top and the bottom by (1 + i):[frac{(1 + i)(1 + i)}{(1 - i)(1 + i)}]Let me expand the numerator first. Multiplying (1 + i) by (1 + i) gives:[1 cdot 1 + 1 cdot i + i cdot 1 + i cdot i = 1 + i + i + i^2]I know that (i^2 = -1), so this simplifies to:[1 + 2i - 1 = 2i]Now, the denominator: multiplying (1 - i) by (1 + i). That's a difference of squares, right?[(1)^2 - (i)^2 = 1 - (-1) = 1 + 1 = 2]So, putting it all together, the fraction simplifies to:[frac{2i}{2} = i]Okay, so the expression simplifies to (i). Now, I need to determine if this is an eighth root of unity. I recall that the eighth roots of unity are the complex numbers that satisfy the equation (z^8 = 1). They can be expressed in the form:[cos frac{2npi}{8} + i sin frac{2npi}{8}]for (n = 0, 1, 2, ..., 7).I know that (i) can be represented in polar form. Since (i) is at 90 degrees or (frac{pi}{2}) radians on the complex plane, its polar form is:[cos frac{pi}{2} + i sin frac{pi}{2}]Comparing this to the general form of the eighth roots of unity, I can set:[frac{2npi}{8} = frac{pi}{2}]Solving for (n):[frac{2npi}{8} = frac{pi}{2} frac{n}{4} = frac{1}{2} n = 2]So, (i) is indeed an eighth root of unity, specifically when (n = 2). Let me double-check this. If I plug (n = 2) into the eighth root formula:[cos frac{4pi}{8} + i sin frac{4pi}{8} = cos frac{pi}{2} + i sin frac{pi}{2} = 0 + i cdot 1 = i]Yes, that matches. So, everything checks out. The expression simplifies to (i), which is the eighth root of unity corresponding to (n = 2).**Final Answer**The value of (n) is boxed{2}.

💡Okay, so I need to solve this problem about an isosceles right triangle with legs of length ( a ). The question asks me to find the value of the expression ( frac{2a}{sqrt{a^2 + a^2}} ) and then verify if this corresponds to the maximum quotient ( frac{a + b}{c} ) in this context. Hmm, let me break this down step by step.First, I know that in an isosceles right triangle, the two legs are equal in length, which are both ( a ) in this case. The hypotenuse ( c ) can be found using the Pythagorean theorem. So, according to the theorem, ( c^2 = a^2 + a^2 ). Let me write that down:( c^2 = a^2 + a^2 )Simplifying the right side, since both terms are ( a^2 ), I get:( c^2 = 2a^2 )To find ( c ), I take the square root of both sides:( c = sqrt{2a^2} )Which simplifies to:( c = asqrt{2} )Okay, so the hypotenuse ( c ) is ( asqrt{2} ). Now, the expression I need to evaluate is ( frac{2a}{sqrt{a^2 + a^2}} ). Let me substitute the value of ( c ) into this expression. Wait, actually, the denominator ( sqrt{a^2 + a^2} ) is the same as the hypotenuse ( c ), right? Because ( sqrt{a^2 + a^2} = c = asqrt{2} ). So, substituting that in, the expression becomes:( frac{2a}{asqrt{2}} )I can simplify this by canceling out the ( a ) in the numerator and the denominator:( frac{2}{sqrt{2}} )Hmm, ( frac{2}{sqrt{2}} ) can be simplified further by rationalizing the denominator. Multiplying numerator and denominator by ( sqrt{2} ):( frac{2 times sqrt{2}}{sqrt{2} times sqrt{2}} = frac{2sqrt{2}}{2} )The 2's cancel out, leaving me with:( sqrt{2} )So, the value of the expression ( frac{2a}{sqrt{a^2 + a^2}} ) is ( sqrt{2} ).Now, the second part of the problem asks me to verify if this corresponds to the maximum quotient ( frac{a + b}{c} ) in this context. Let me think about this. In an isosceles right triangle, both legs are equal, so ( a = b ). Therefore, ( a + b = 2a ). The hypotenuse ( c ) we already found is ( asqrt{2} ). So, the quotient ( frac{a + b}{c} ) becomes:( frac{2a}{asqrt{2}} = frac{2}{sqrt{2}} = sqrt{2} )Wait a minute, that's the same result as before. So, in this specific case, the quotient ( frac{a + b}{c} ) equals ( sqrt{2} ). But the question is asking if this is the maximum quotient. Hmm, is ( sqrt{2} ) the maximum possible value for ( frac{a + b}{c} ) in an isosceles right triangle?Let me consider if there could be a scenario where ( frac{a + b}{c} ) is larger than ( sqrt{2} ). Since in an isosceles right triangle, the legs are equal, and the hypotenuse is fixed based on the legs, I don't think ( frac{a + b}{c} ) can be larger than ( sqrt{2} ) here. If the triangle weren't isosceles, maybe the quotient could be different, but since it's specifically isosceles, the maximum is achieved when both legs are equal, which gives us ( sqrt{2} ).Therefore, I conclude that ( frac{2a}{sqrt{a^2 + a^2}} = sqrt{2} ) and this is indeed the maximum value of ( frac{a + b}{c} ) for an isosceles right triangle with legs of length ( a ).

💡First, I need to determine the number of ways to choose 4 boxes out of the 6 to place the digits 1, 2, 3, and 4. This can be done using the combination formula, which is ( binom{6}{4} ).Next, for each selection of 4 boxes, I need to calculate the number of ways to arrange the 4 distinct digits. Since each digit must be placed in a separate box, this is a permutation problem, and the number of permutations is ( 4! ).Finally, the two remaining boxes will be left blank, and there is only 1 way to do this.By multiplying the number of ways to choose the boxes by the number of ways to arrange the digits, I can find the total number of distinct arrangements.

💡Okay, so I have this problem here that says Ben was given 7 trays of eggs to examine, and each tray has 10 eggs. The question is asking how many eggs Ben examined in total. Hmm, let's see. I think this is a multiplication problem because we're dealing with groups of items. Each tray is a group, and each group has 10 eggs. So, if there are 7 trays, I need to find out the total number of eggs by multiplying the number of trays by the number of eggs in each tray.Wait, but let me make sure I understand the problem correctly. It says Ben was "also" given 7 trays, which makes me think maybe there was something else mentioned before this, like maybe he had some other eggs or trays to examine too. But in the problem as it's presented, it only mentions these 7 trays with 10 eggs each. So, I guess I can focus just on these 7 trays for now.Alright, so if each tray has 10 eggs, and there are 7 trays, then the total number of eggs would be 7 multiplied by 10. Let me write that down: 7 times 10 equals... hmm, 70. Yeah, that seems right. But wait, is there any chance that some eggs might be broken or not counted? The problem doesn't mention anything like that, so I think it's safe to assume all eggs are intact and need to be examined.Also, the problem says Ben was given these trays to "examine." I wonder if examining means he just looked at them or if he had to do something more, like count them or check for quality. But regardless of what examining entails, the question is only asking about the total number of eggs, not about the process of examining them. So, I don't need to worry about that part.Let me double-check my multiplication to be sure. 7 times 10 is 70. Yep, that's correct. I could also think of it as adding 10 seven times: 10 plus 10 is 20, plus another 10 is 30, and so on until I reach 70. That matches what I got with multiplication.Is there any other way to approach this problem? Maybe using division or addition? Well, addition would work if I add 10 seven times, but that's essentially the same as multiplying 7 by 10. Division doesn't seem necessary here because we're not splitting anything up; we're combining groups.So, in conclusion, Ben examined a total of 70 eggs. That makes sense based on the information given. I don't think I've missed anything here. The problem is straightforward, and the calculation seems correct.