answer：To find the coefficient of ( x^3y^3 ) in the expansion of ( (x + y)^4 ), we can use the binomial theorem. The binomial theorem states that:[(a + b)^n = sum_{k=0}^{n} binom{n}{k} a^{n - k} b^k]Here, ( a = x ), ( b = y ), and ( n = 4 ). We are interested in the term where the powers of ( x ) and ( y ) are both 3. This corresponds to ( k = 3 ) in the binomial expansion.The binomial coefficient for this term is:[binom{4}{3} = frac{4!}{3!(4 - 3)!} = frac{24}{6 times 1} = 4]Therefore, the coefficient of ( x^3y^3 ) in the expansion of ( (x + y)^4 ) is 4.

answer：First, I need to evaluate the expression (201 times 5 + 1220 - 2 times 3 times 5 times 7).I'll start by calculating (201 times 5). Breaking it down, (201 times 5) is the same as (200 times 5 + 1 times 5), which equals (1000 + 5 = 1005).Next, I'll simplify the multiplication part of the expression, (2 times 3 times 5 times 7). I can rearrange the multiplication for easier calculation: (2 times 5 times 3 times 7). This simplifies to ((2 times 5) times (3 times 7) = 10 times 21 = 210).Now, I'll substitute these values back into the original expression: (1005 + 1220 - 210).Adding (1005) and (1220) gives me (2225). Then, subtracting (210) from (2225) results in (2015).Therefore, the final answer is (2015).

answer：Alright, so I have this problem here with a circle and a parabola, and I need to find the equation of a line that passes through the center of the circle and intersects both curves at four points. The segments between these points form an arithmetic sequence. Hmm, okay, let me try to break this down step by step.First, let's look at the circle P: x² + y² - 4y = 0. I remember that the general equation of a circle is x² + y² + Dx + Ey + F = 0, and its center is at (-D/2, -E/2). So, for this circle, D is 0 and E is -4. That means the center is at (0, 2). Got that. The radius can be found using the formula r = √(D² + E² - 4F)/2. Plugging in the values, D is 0, E is -4, and F is 0, so r = √(0 + 16 - 0)/2 = √16/2 = 4/2 = 2. So, the circle has a radius of 2 and is centered at (0, 2). Cool.Next, the parabola S is given by y = x²/8. That's a standard upward-opening parabola with vertex at the origin. The coefficient 1/8 tells me it's wider than the standard parabola y = x². So, it's spread out more along the x-axis.Now, the line l passes through the center of the circle, which is (0, 2). So, the line must pass through this point. Let's denote the equation of line l as y = kx + 2, where k is the slope. That makes sense because when x is 0, y is 2, which is the center.This line intersects both the circle and the parabola. Since it's passing through the center of the circle, it will intersect the circle at two points, and it will intersect the parabola at two other points. So, in total, we have four intersection points: A, B, C, D, from left to right.The problem states that the lengths of segments AB, BC, and CD form an arithmetic sequence. That means AB, BC, CD are in an arithmetic progression. So, if AB = a, BC = a + d, and CD = a + 2d, but since it's an arithmetic sequence, the difference between consecutive terms is constant. Alternatively, it could be that AB, BC, CD are equally spaced, but I think the key is that the differences are equal.Wait, actually, an arithmetic sequence means that the difference between consecutive terms is constant. So, BC - AB = CD - BC, which implies that 2BC = AB + CD. That's a useful relationship.So, let's denote AB = x, BC = x + d, CD = x + 2d. But since it's an arithmetic sequence, the middle term BC is the average of AB and CD. So, BC = (AB + CD)/2. That might be a useful way to think about it.But before getting too deep into that, maybe I should find the points of intersection between the line and the circle, and between the line and the parabola.Starting with the circle: x² + y² - 4y = 0. Since the line is y = kx + 2, I can substitute this into the circle's equation.Substituting y = kx + 2 into x² + y² - 4y = 0:x² + (kx + 2)² - 4(kx + 2) = 0Let me expand that:x² + (k²x² + 4kx + 4) - 4kx - 8 = 0Simplify term by term:x² + k²x² + 4kx + 4 - 4kx - 8 = 0Combine like terms:(1 + k²)x² + (4kx - 4kx) + (4 - 8) = 0So, the 4kx and -4kx cancel out, and 4 - 8 is -4.Thus, we have (1 + k²)x² - 4 = 0So, (1 + k²)x² = 4Therefore, x² = 4 / (1 + k²)So, x = ±2 / √(1 + k²)Therefore, the points of intersection with the circle are at x = 2 / √(1 + k²) and x = -2 / √(1 + k²). Let's denote these as points B and C, since they are the intersections with the circle, which is centered at (0, 2). So, from left to right, the points would be A, B, C, D.Wait, actually, the line intersects the parabola at two points and the circle at two points. So, depending on the slope, the order could vary. But since the parabola is opening upwards and the circle is centered at (0, 2), the intersections with the parabola will be further out on the x-axis compared to the circle.Therefore, from left to right, the points should be A (leftmost intersection with parabola), then B (left intersection with circle), then C (right intersection with circle), then D (rightmost intersection with parabola). So, A, B, C, D.So, let's find the points of intersection with the parabola as well. The parabola is y = x² / 8, and the line is y = kx + 2. So, set them equal:x² / 8 = kx + 2Multiply both sides by 8:x² = 8kx + 16Bring all terms to one side:x² - 8kx - 16 = 0This is a quadratic equation in x. Let's denote the solutions as x1 and x2.Using the quadratic formula:x = [8k ± √(64k² + 64)] / 2Simplify:x = [8k ± 8√(k² + 1)] / 2x = 4k ± 4√(k² + 1)So, the x-coordinates of points A and D are 4k - 4√(k² + 1) and 4k + 4√(k² + 1), respectively.Therefore, the coordinates of A and D are:A: (4k - 4√(k² + 1), y1)D: (4k + 4√(k² + 1), y2)But since y = kx + 2, we can write y1 = k*(4k - 4√(k² + 1)) + 2 and y2 = k*(4k + 4√(k² + 1)) + 2.But maybe we don't need the exact y-coordinates right now.Now, let's think about the distances AB, BC, and CD.Since all points lie on the line y = kx + 2, the distance between two points can be found using the distance formula along the line, which simplifies to the absolute difference in x-coordinates multiplied by the square root of (1 + k²). Because the distance between two points (x1, y1) and (x2, y2) on a line with slope k is |x2 - x1| * √(1 + k²).Wait, actually, the distance formula is √[(x2 - x1)² + (y2 - y1)²]. Since y = kx + 2, y2 - y1 = k(x2 - x1). So, the distance becomes √[(x2 - x1)² + (k(x2 - x1))²] = |x2 - x1| * √(1 + k²). So, yes, the distance between two points on the line is |x2 - x1| * √(1 + k²).Therefore, AB, BC, and CD can be expressed in terms of the differences in x-coordinates multiplied by √(1 + k²).So, let's denote:AB = |xB - xA| * √(1 + k²)BC = |xC - xB| * √(1 + k²)CD = |xD - xC| * √(1 + k²)Since the points are ordered from left to right as A, B, C, D, the x-coordinates satisfy xA < xB < xC < xD. Therefore, the differences are positive, so we can drop the absolute value:AB = (xB - xA) * √(1 + k²)BC = (xC - xB) * √(1 + k²)CD = (xD - xC) * √(1 + k²)Given that AB, BC, CD form an arithmetic sequence, we have:2BC = AB + CDSo, 2*(xC - xB) = (xB - xA) + (xD - xC)Simplify:2xC - 2xB = xB - xA + xD - xCBring all terms to one side:2xC - 2xB - xB + xA - xD + xC = 0Combine like terms:(2xC + xC) + (-2xB - xB) + xA - xD = 03xC - 3xB + xA - xD = 0Divide both sides by 3:xC - xB + (xA - xD)/3 = 0Hmm, not sure if that helps directly. Maybe another approach.Alternatively, since AB, BC, CD are in arithmetic sequence, the differences between consecutive terms are equal. So, BC - AB = CD - BC, which implies 2BC = AB + CD, as before.But perhaps instead of dealing with the distances directly, we can think about the x-coordinates.Let me denote the x-coordinates as follows:xA: leftmost intersection with parabolaxB: left intersection with circlexC: right intersection with circlexD: rightmost intersection with parabolaWe have the x-coordinates for the circle intersections: xB = -2 / √(1 + k²) and xC = 2 / √(1 + k²)Wait, earlier I found that the x-coordinates for the circle intersections are ±2 / √(1 + k²). So, xB = -2 / √(1 + k²) and xC = 2 / √(1 + k²)And for the parabola intersections, xA = 4k - 4√(k² + 1) and xD = 4k + 4√(k² + 1)So, now, let's write down the differences:AB = xB - xA = [-2 / √(1 + k²)] - [4k - 4√(k² + 1)]BC = xC - xB = [2 / √(1 + k²)] - [-2 / √(1 + k²)] = 4 / √(1 + k²)CD = xD - xC = [4k + 4√(k² + 1)] - [2 / √(1 + k²)]But remember, these differences are multiplied by √(1 + k²) to get the actual distances AB, BC, CD.So, let's compute AB, BC, CD:AB = [xB - xA] * √(1 + k²) = [(-2 / √(1 + k²)) - (4k - 4√(k² + 1))] * √(1 + k²)Simplify:= [ -2 / √(1 + k²) - 4k + 4√(k² + 1) ] * √(1 + k²)= -2 - 4k√(1 + k²) + 4(k² + 1)Similarly, BC = [xC - xB] * √(1 + k²) = [4 / √(1 + k²)] * √(1 + k²) = 4And CD = [xD - xC] * √(1 + k²) = [4k + 4√(k² + 1) - 2 / √(1 + k²)] * √(1 + k²)Simplify:= [4k√(1 + k²) + 4(k² + 1) - 2]So, now we have expressions for AB, BC, CD.Given that AB, BC, CD form an arithmetic sequence, so 2BC = AB + CD.We know BC = 4, so 2*4 = 8 = AB + CD.So, AB + CD = 8.Let me compute AB + CD:AB + CD = [ -2 - 4k√(1 + k²) + 4(k² + 1) ] + [4k√(1 + k²) + 4(k² + 1) - 2 ]Simplify term by term:-2 - 4k√(1 + k²) + 4k² + 4 + 4k√(1 + k²) + 4k² + 4 - 2Combine like terms:-2 + 4 + 4 - 2 = 4-4k√(1 + k²) + 4k√(1 + k²) = 04k² + 4k² = 8k²So, AB + CD = 8k² + 4But we know AB + CD = 8, so:8k² + 4 = 8Subtract 4:8k² = 4Divide by 8:k² = 4 / 8 = 1/2Therefore, k = ±√(1/2) = ±√2 / 2So, the slopes are √2 / 2 and -√2 / 2Therefore, the equations of the line l are:y = (√2 / 2)x + 2andy = (-√2 / 2)x + 2Looking back at the options, option B is y = -√2/2 x + 2 or y = √2/2 x + 2, which matches our result.Wait, let me double-check my calculations to make sure I didn't make a mistake.Starting from AB + CD = 8k² + 4 = 8, leading to k² = 1/2. That seems correct.Yes, so k = ±√(1/2) = ±√2 / 2.So, the equations are y = ±(√2 / 2)x + 2, which is option B.I think that makes sense. The line passes through the center of the circle, intersects both the circle and parabola, and the distances between the intersection points form an arithmetic sequence. The key was setting up the equation AB + CD = 2BC, which led us to solve for k.I don't see any errors in the steps, so I'm confident that option B is the correct answer.

answer：To find the value of sin alpha, we start by determining the slope of the given line x - 2y + 2 = 0. 1. **Rewrite the equation in slope-intercept form (y = mx + b):** [ x - 2y + 2 = 0 implies -2y = -x - 2 implies y = frac{1}{2}x + 1 ] The slope (m) of the line is frac{1}{2}.2. **Since line l is parallel to this line, it has the same slope:** [ tan alpha = frac{1}{2} ]3. **Use the trigonometric identity to find sin alpha:** [ sin alpha = frac{tan alpha}{sqrt{1 + tan^2 alpha}} = frac{frac{1}{2}}{sqrt{1 + left(frac{1}{2}right)^2}} = frac{frac{1}{2}}{sqrt{frac{5}{4}}} = frac{frac{1}{2}}{frac{sqrt{5}}{2}} = frac{1}{sqrt{5}} = frac{sqrt{5}}{5} ]Therefore, the value of sin alpha is frac{sqrt{5}}{5}.

answer：First, I recognize that there are two division scenarios: one where the student used the incorrect divisor and another where the correct divisor of 36 was used.Let’s denote the dividend as ( D ) and the incorrect divisor as ( x ).From the student's incorrect division:[frac{D}{x} = 24]This implies:[D = 24x]From the correct division:[frac{D}{36} = 48]This implies:[D = 48 times 36]Since both expressions equal ( D ), I can set them equal to each other:[24x = 48 times 36]Solving for ( x ):[x = frac{48 times 36}{24} = 2 times 36 = 72]Therefore, the incorrect divisor the student used was 72.

answer：Alright, so I've got this problem here about a school organizing a trip for 540 eighth-grade students. They need to rent buses to transport everyone, and there are two types of buses: type A and type B. The problem has two parts. The first part is about figuring out how many seats each type of bus has, and the second part is about minimizing the rental cost while ensuring all students and teachers can go on the trip.Starting with part (1), I need to find the number of passenger seats in each type A and type B bus. The problem says that if the school rents 8 type A buses and 4 type B buses, all the buses will be fully occupied. Also, each type A bus has 12 more seats than each type B bus.Okay, so let's denote the number of seats in each type B bus as x. Then, the number of seats in each type A bus would be x + 12. Since there are 8 type A buses and 4 type B buses, the total number of seats would be 8*(x + 12) + 4*x. And this total should equal the number of students, which is 540.So, setting up the equation: 8*(x + 12) + 4*x = 540.Let me compute that step by step. First, expand the terms: 8*x + 8*12 + 4*x = 540.Calculating 8*12 gives 96. So, the equation becomes 8x + 96 + 4x = 540.Combining like terms, 8x + 4x is 12x, so 12x + 96 = 540.Now, subtract 96 from both sides: 12x = 540 - 96.540 minus 96 is 444, so 12x = 444.Dividing both sides by 12, x = 444 / 12.Calculating that, 444 divided by 12 is 37. So, x = 37.That means each type B bus has 37 seats. Then, each type A bus has x + 12, which is 37 + 12 = 49 seats.Alright, so part (1) seems solved: type A buses have 49 seats each, and type B buses have 37 seats each.Moving on to part (2). The school wants to adjust the rental plan to minimize the total cost. They mention that one teacher must accompany each bus, so the number of teachers is equal to the number of buses. The total number of people to transport is now 540 students plus the number of teachers, which is the same as the number of buses.The original plan was 8 type A and 4 type B buses, totaling 12 buses. So, the number of teachers would be 12, making the total number of people 540 + 12 = 552.The cost to rent a type A bus is 2100 yuan, and type B is 1500 yuan. The school wants to keep the total number of buses unchanged, which is 12 buses, but adjust the number of type A and type B buses to minimize the total cost.So, we need to find how many type A buses (let's call this number m) and type B buses (which would be 12 - m) should be rented so that the total cost is minimized, while ensuring that all 552 people can be transported.First, let's set up the inequality for the total seating capacity. Each type A bus has 49 seats, and each type B has 37 seats. The total number of seats must be at least 552.So, the inequality is: 49*m + 37*(12 - m) >= 552.Let me compute this step by step. Expanding the terms: 49m + 37*12 - 37m >= 552.Calculating 37*12: 37*10 is 370, and 37*2 is 74, so 370 + 74 = 444.So, the inequality becomes: 49m + 444 - 37m >= 552.Combining like terms: 49m - 37m = 12m, so 12m + 444 >= 552.Subtracting 444 from both sides: 12m >= 552 - 444.552 minus 444 is 108, so 12m >= 108.Dividing both sides by 12: m >= 9.So, m must be at least 9. Since the total number of buses is 12, m can be from 9 to 12.Now, to minimize the cost, we need to find the combination of m and (12 - m) that gives the lowest total cost.The total cost is given by: 2100*m + 1500*(12 - m).Let's express this as a function: w = 2100m + 1500*(12 - m).Expanding this: w = 2100m + 18000 - 1500m.Combining like terms: 2100m - 1500m = 600m, so w = 600m + 18000.Now, since the coefficient of m is positive (600), the total cost increases as m increases. Therefore, to minimize the cost, we should choose the smallest possible value of m, which is 9.So, if m = 9, then the number of type B buses is 12 - 9 = 3.Calculating the total cost: w = 600*9 + 18000.600*9 is 5400, so 5400 + 18000 = 23400 yuan.Therefore, renting 9 type A buses and 3 type B buses will minimize the total cost, which is 23400 yuan.Let me just double-check to make sure this makes sense. If we rent 9 type A buses, each with 49 seats, that's 9*49 = 441 seats. 3 type B buses, each with 37 seats, is 3*37 = 111 seats. Total seats: 441 + 111 = 552, which matches the total number of people (540 students + 12 teachers). So, the seating is exactly enough, and the cost is minimized because we're using as few type A buses (which are more expensive) as possible while still meeting the seating requirement.Yeah, that seems correct.