answer：First, I'll assign variables to represent the ages of A, B, and C.Let A be the age of person A, B be the age of person B, and C be the age of person C.According to the problem, the total age of A and B is 12 years more than the total age of B and C. This can be written as:A + B = B + C + 12To find out how much younger C is than A, I'll rearrange the equation to isolate C in terms of A.Subtract B from both sides:A = C + 12This shows that C is 12 years younger than A.Therefore, C is 12 years younger than A.

answer：First, I need to determine the total number of ways to select 3 pencils out of the 10 available. This can be calculated using the combination formula, which is:[text{Total ways} = binom{10}{3} = frac{10!}{3!(10-3)!} = frac{10 times 9 times 8}{3 times 2 times 1} = 120]Next, I need to find the number of ways to select 3 non-defective pencils out of the 8 non-defective ones. Using the combination formula again:[text{Ways to select non-defective pencils} = binom{8}{3} = frac{8!}{3!(8-3)!} = frac{8 times 7 times 6}{3 times 2 times 1} = 56]Finally, to find the probability that none of the selected pencils are defective, I divide the number of ways to select 3 non-defective pencils by the total number of ways to select any 3 pencils:[text{Probability} = frac{56}{120} = frac{7}{15}]

answer：Alright, so I have this problem about an arithmetic sequence. Let me try to understand it step by step. First, the problem says that {a_n} is an arithmetic sequence. I remember that in an arithmetic sequence, each term is obtained by adding a constant difference, d, to the previous term. So, the general form is a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.Next, it defines another sequence {b_n} where each term is the product of three consecutive terms of {a_n}. Specifically, b_n = a_n * a_{n+1} * a_{n+2}. So, for each n, b_n is the product of a_n, the next term, and the one after that.Then, S_n is the sum of the first n terms of {b_n}. So, S_n = b_1 + b_2 + ... + b_n. The problem asks for the value of n when S_n reaches its maximum.We are also given that 3a_5 = 8a_{12} > 0. So, this is a condition that relates the 5th and 12th terms of the arithmetic sequence. Since it's given that this product is greater than zero, it tells us something about the signs of these terms.Let me start by using the given condition: 3a_5 = 8a_{12} > 0.Since {a_n} is an arithmetic sequence, I can express a_5 and a_{12} in terms of a_1 and d.a_5 = a_1 + 4da_{12} = a_1 + 11dSo, substituting these into the equation:3(a_1 + 4d) = 8(a_1 + 11d)Let me expand both sides:3a_1 + 12d = 8a_1 + 88dNow, let's bring all terms to one side:3a_1 + 12d - 8a_1 - 88d = 0Simplify:-5a_1 - 76d = 0So, -5a_1 = 76dTherefore, a_1 = (-76/5)dHmm, so the first term is expressed in terms of the common difference d.Also, since 3a_5 = 8a_{12} > 0, this tells us that both a_5 and a_{12} are positive because their product is positive. Wait, actually, it's 3a_5 = 8a_{12}, and both sides are positive because their product is positive. So, a_5 and a_{12} must be positive.But since it's an arithmetic sequence, the terms can be increasing or decreasing depending on the common difference d.Given that a_5 and a_{12} are both positive, but let's see if the sequence is increasing or decreasing.From the equation a_1 = (-76/5)d, so if d is positive, then a_1 is negative. If d is negative, then a_1 is positive.But since a_5 and a_{12} are positive, let's see what that implies.Let me compute a_5:a_5 = a_1 + 4d = (-76/5)d + 4d = (-76/5 + 20/5)d = (-56/5)dSimilarly, a_{12} = a_1 + 11d = (-76/5)d + 11d = (-76/5 + 55/5)d = (-21/5)dBut we know that 3a_5 = 8a_{12} > 0, so both a_5 and a_{12} must be positive.So, (-56/5)d > 0 and (-21/5)d > 0.This implies that d must be negative because multiplying a negative d by a negative coefficient gives a positive result.So, d < 0.Therefore, the common difference is negative, meaning the arithmetic sequence is decreasing.So, starting from a positive a_1, each subsequent term decreases by |d|.Given that, let's see when the terms of {a_n} become negative.We can find the term where a_n changes sign from positive to negative.Since a_n = a_1 + (n-1)dWe can set a_n = 0 to find the critical point.a_n = 0 => a_1 + (n-1)d = 0We know a_1 = (-76/5)d, so:(-76/5)d + (n - 1)d = 0Factor out d:[(-76/5) + (n - 1)]d = 0Since d ≠ 0, we have:(-76/5) + (n - 1) = 0So, n - 1 = 76/5n = 76/5 + 1 = 81/5 = 16.2So, the 16.2th term is zero. Since n must be an integer, the 16th term is the last positive term, and the 17th term is the first negative term.Therefore, a_{16} > 0 and a_{17} < 0.So, the sequence {a_n} is positive up to n=16 and negative from n=17 onwards.Now, let's think about the sequence {b_n} = a_n * a_{n+1} * a_{n+2}.Since {a_n} is positive up to n=16 and negative from n=17 onwards, let's see the sign of each b_n.For n <=14:a_n, a_{n+1}, a_{n+2} are all positive because n+2 <=16. So, b_n is positive.For n=15:a_{15}, a_{16}, a_{17}. Here, a_{15} and a_{16} are positive, but a_{17} is negative. So, the product b_{15} is negative.For n=16:a_{16}, a_{17}, a_{18}. Here, a_{16} is positive, a_{17} and a_{18} are negative. So, the product b_{16} is positive because positive * negative * negative = positive.For n >=17:a_n, a_{n+1}, a_{n+2} are all negative or mixed, but since the sequence is decreasing, let's see:Wait, actually, from n=17 onwards, all terms are negative, so the product of three negative terms is negative. So, b_n for n >=17 is negative.So, the sequence {b_n} is positive for n=1 to n=14, negative for n=15, positive for n=16, and negative for n>=17.But wait, let me check n=16 again.a_{16} is positive, a_{17} is negative, a_{18} is negative. So, positive * negative * negative = positive. So, b_{16} is positive.Similarly, for n=17: a_{17}, a_{18}, a_{19} are all negative, so their product is negative.So, the signs of b_n are:n=1: positive...n=14: positiven=15: negativen=16: positiven=17: negativen=18: negative...So, the sequence {b_n} has positive terms up to n=14, then negative at n=15, positive at n=16, and then negative onwards.Now, we need to find when the sum S_n reaches its maximum.Since S_n is the sum of b_1 to b_n, and b_n changes sign, the maximum sum will occur before the sum starts decreasing.But let's analyze the behavior of S_n.From n=1 to n=14, each b_n is positive, so S_n is increasing.At n=15, b_{15} is negative, so S_{15} = S_{14} + b_{15}. Since b_{15} is negative, S_{15} is less than S_{14}.At n=16, b_{16} is positive, so S_{16} = S_{15} + b_{16}. Since b_{16} is positive, S_{16} is greater than S_{15}.But we need to check if S_{16} is greater than S_{14}.Because S_{16} = S_{14} + b_{15} + b_{16}If b_{15} + b_{16} > 0, then S_{16} > S_{14}If b_{15} + b_{16} < 0, then S_{16} < S_{14}So, we need to compute b_{15} and b_{16} to see if their sum is positive or negative.Let me compute b_{15} and b_{16}.First, let's find expressions for a_n.We have a_n = a_1 + (n-1)dWe know a_1 = (-76/5)dSo, a_n = (-76/5)d + (n-1)d = [(-76/5) + (n-1)]dLet me compute a_{15}, a_{16}, a_{17}, a_{18}.a_{15} = [(-76/5) + 14]d = [(-76/5) + (70/5)]d = (-6/5)da_{16} = [(-76/5) + 15]d = [(-76/5) + (75/5)]d = (-1/5)da_{17} = [(-76/5) + 16]d = [(-76/5) + (80/5)]d = (4/5)dWait, but earlier we concluded that a_{17} is negative because d < 0.Wait, let's check:a_{17} = a_1 + 16d = (-76/5)d + 16d = (-76/5 + 80/5)d = (4/5)dBut since d < 0, a_{17} = (4/5)d < 0.Similarly, a_{18} = a_1 + 17d = (-76/5)d + 17d = (-76/5 + 85/5)d = (9/5)d < 0So, a_{15} = (-6/5)d > 0 because d < 0a_{16} = (-1/5)d > 0a_{17} = (4/5)d < 0a_{18} = (9/5)d < 0Now, compute b_{15} = a_{15} * a_{16} * a_{17}= [(-6/5)d] * [(-1/5)d] * [(4/5)d]Multiply the coefficients:(-6/5) * (-1/5) * (4/5) = (6/25) * (4/5) = 24/125Multiply by d^3:b_{15} = (24/125)d^3But since d < 0, d^3 < 0, so b_{15} < 0Similarly, compute b_{16} = a_{16} * a_{17} * a_{18}= [(-1/5)d] * [(4/5)d] * [(9/5)d]Multiply the coefficients:(-1/5) * (4/5) * (9/5) = (-36/125)Multiply by d^3:b_{16} = (-36/125)d^3But d < 0, so d^3 < 0, so b_{16} = (-36/125)d^3 > 0 because negative times negative is positive.So, b_{15} = (24/125)d^3 < 0b_{16} = (-36/125)d^3 > 0Now, let's compute b_{15} + b_{16}:(24/125)d^3 + (-36/125)d^3 = (-12/125)d^3But d < 0, so d^3 < 0, so (-12/125)d^3 > 0Wait, that can't be right. Let me check the signs again.Wait, b_{15} is (24/125)d^3, but d < 0, so d^3 < 0, so b_{15} = (24/125)(negative) = negative.b_{16} is (-36/125)d^3, which is (-36/125)(negative) = positive.So, b_{15} + b_{16} = negative + positive.We need to see if the positive term is larger in magnitude than the negative term.Compute the absolute values:|b_{15}| = |24/125| * |d|^3 = (24/125)|d|^3|b_{16}| = | -36/125 | * |d|^3 = (36/125)|d|^3So, |b_{16}| > |b_{15}|, so b_{15} + b_{16} is positive because the positive term is larger.Therefore, S_{16} = S_{14} + b_{15} + b_{16} > S_{14} because b_{15} + b_{16} > 0.So, S_{16} is greater than S_{14}.Now, let's check S_{17}:S_{17} = S_{16} + b_{17}But b_{17} = a_{17} * a_{18} * a_{19}Compute a_{19} = a_1 + 18d = (-76/5)d + 18d = (-76/5 + 90/5)d = (14/5)d < 0So, b_{17} = a_{17} * a_{18} * a_{19} = (4/5)d * (9/5)d * (14/5)d = (4*9*14)/(5^3) d^3 = (504/125)d^3Since d < 0, d^3 < 0, so b_{17} = (504/125)(negative) = negative.So, b_{17} is negative, so S_{17} = S_{16} + b_{17} < S_{16}Similarly, for n >=17, b_n is negative, so adding them will decrease S_n.Therefore, the maximum sum occurs at n=16.Wait, but let me confirm this.We have S_{14} is the sum up to b_{14}, which is all positive terms.Then, S_{15} = S_{14} + b_{15}, which is less than S_{14} because b_{15} is negative.But then S_{16} = S_{15} + b_{16}, which is greater than S_{15} because b_{16} is positive, and as we saw, b_{15} + b_{16} > 0, so S_{16} > S_{14}.Then, S_{17} = S_{16} + b_{17}, which is less than S_{16} because b_{17} is negative.So, the maximum occurs at n=16.Therefore, the value of n when S_n reaches its maximum is 16.

answer：First, I recognize that the side of the square is increased by 10%. This means the new side length is 1.10 times the original side length.Next, I calculate the new area of the square by squaring the new side length. This gives me (1.10s)², which simplifies to 1.21s².To find the percentage increase in the area, I compare the new area to the original area. The difference between the new area and the original area is 1.21s² - s² = 0.21s².Finally, I determine the percentage increase by dividing the difference by the original area and multiplying by 100%. This results in (0.21s² / s²) × 100% = 21%.

answer：First, I need to determine the distance from the point ( P(1, 2, 2) ) to the origin in a three-dimensional space. I recall that the distance formula in three dimensions is given by:[text{Distance} = sqrt{x^2 + y^2 + z^2}]Applying this formula to the coordinates of point ( P ):[text{Distance} = sqrt{1^2 + 2^2 + 2^2} = sqrt{1 + 4 + 4} = sqrt{9} = 3]Therefore, the distance from point ( P(1, 2, 2) ) to the origin is 3.

answer：Okay, so I need to find the area of a triangle with vertices at (4, -1), (10, 3), and (4, 5). Hmm, let's see. I remember there are a few ways to calculate the area of a triangle when you know the coordinates of its vertices. One common method is using the formula that involves the coordinates directly. I think it's something like taking the determinant of a matrix or using the shoelace formula. Wait, what's the exact formula again?Oh yeah, the shoelace formula! It's a way to find the area of any polygon when you know the coordinates of its vertices. For a triangle, it should be straightforward. The formula is:Area = (1/2) * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|Let me write down the coordinates again to make sure I have them right:- Point A: (4, -1)- Point B: (10, 3)- Point C: (4, 5)So, plugging these into the formula:x₁ = 4, y₁ = -1x₂ = 10, y₂ = 3x₃ = 4, y₃ = 5Now, let's compute each part step by step.First part: x₁(y₂ - y₃) = 4*(3 - 5) = 4*(-2) = -8Second part: x₂(y₃ - y₁) = 10*(5 - (-1)) = 10*(6) = 60Third part: x₃(y₁ - y₂) = 4*(-1 - 3) = 4*(-4) = -16Now, add these three parts together:-8 + 60 + (-16) = (-8 -16) + 60 = (-24) + 60 = 36Take the absolute value (which is still 36) and multiply by 1/2:Area = (1/2)*36 = 18Wait, that seems straightforward. But just to double-check, maybe I can visualize the triangle or use another method to confirm.Looking at the points, I notice that two of them have the same x-coordinate: (4, -1) and (4, 5). That means they lie on a vertical line. So, the distance between these two points is just the difference in their y-coordinates: 5 - (-1) = 6 units. That could be the base of the triangle.Now, the third point is (10, 3). To find the height, I need the horizontal distance from this point to the vertical line x=4. Since the x-coordinate of the third point is 10, the distance is 10 - 4 = 6 units.So, using the basic area formula for a triangle: (1/2)*base*height = (1/2)*6*6 = 18. That matches what I got earlier with the shoelace formula. Good, so both methods give the same result.Just to make sure I didn't make any calculation errors, let me go through the shoelace formula again.Compute x₁(y₂ - y₃): 4*(3 - 5) = 4*(-2) = -8Compute x₂(y₃ - y₁): 10*(5 - (-1)) = 10*6 = 60Compute x₃(y₁ - y₂): 4*(-1 - 3) = 4*(-4) = -16Add them up: -8 + 60 -16 = 36Take half of that: 18Yep, still 18. Okay, I feel confident that the area is 18 square units.