answer：First, I need to determine how many ways there are to select exactly two of the triplets (Bob, Bill, and Ben) for the starting lineup. Since there are three triplets and I need to choose two, I can use the combination formula to calculate this.Next, I need to select exactly one of the twins (Tim and Tom). There are two twins, and I need to choose one, so I'll use the combination formula again for this selection.After selecting the required triplets and twins, I need to choose the remaining players from the rest of the team. The total number of players is 18, and I've already selected 2 triplets and 1 twin, leaving 15 players. I need to choose 4 more players from these 15.Finally, I'll multiply the number of ways to choose the triplets, the twins, and the remaining players to find the total number of possible starting lineups.

answer：First, I need to determine the number of matches Rahul has played before today's match. Let's denote this number as ( n ).Rahul's current batting average is 52, which means the total runs he has scored before today's match can be calculated as ( 52n ).If Rahul scores 78 runs in today's match, his total runs will increase to ( 52n + 78 ).After scoring 78 runs, his batting average becomes 54. This means that the total runs scored after today's match divided by the total number of matches played (which is ( n + 1 )) equals 54.I can set up the equation:[frac{52n + 78}{n + 1} = 54]To solve for ( n ), I'll multiply both sides by ( n + 1 ):[52n + 78 = 54(n + 1)]Expanding the right side:[52n + 78 = 54n + 54]Subtracting ( 52n ) from both sides:[78 = 2n + 54]Subtracting 54 from both sides:[24 = 2n]Finally, dividing both sides by 2:[n = 12]Therefore, Rahul has played 12 matches before today's match.

answer：First, I recognize that I need to find the value of (tan(alpha - beta)) given that (tanalpha = 2) and (tanbeta = frac{3}{4}).I recall the formula for the tangent of the difference of two angles:[tan(alpha - beta) = frac{tanalpha - tanbeta}{1 + tanalpha tanbeta}]Next, I substitute the given values into the formula:[tan(alpha - beta) = frac{2 - frac{3}{4}}{1 + 2 times frac{3}{4}}]I then simplify the numerator and the denominator:[text{Numerator: } 2 - frac{3}{4} = frac{8}{4} - frac{3}{4} = frac{5}{4}][text{Denominator: } 1 + 2 times frac{3}{4} = 1 + frac{6}{4} = frac{10}{4} = frac{5}{2}]Finally, I divide the simplified numerator by the simplified denominator to find the value of (tan(alpha - beta)):[tan(alpha - beta) = frac{frac{5}{4}}{frac{5}{2}} = frac{5}{4} times frac{2}{5} = frac{10}{20} = frac{1}{2}]

answer：First, I need to determine how many eggs Joe found around the club house. I know the total number of eggs he found is 20.He found 5 eggs around the park and 3 eggs in the town hall garden. Adding these together gives me 5 + 3 = 8 eggs.To find out how many eggs he found around the club house, I subtract the eggs found in the other locations from the total number of eggs: 20 - 8 = 12.Therefore, Joe found 12 eggs around the club house.

answer：Okay, so I have this problem where the terminal side of angle α passes through the point P(4/5, -3/5). I need to find sin α and then evaluate this trigonometric expression. Hmm, let me start with part (1).First, I remember that for any point (x, y) on the terminal side of an angle in standard position, the sine of the angle is y divided by the radius r, where r is the distance from the origin to the point. So, I need to find r.Given x = 4/5 and y = -3/5, I can calculate r using the Pythagorean theorem:r = sqrt(x² + y²) = sqrt((4/5)² + (-3/5)²) = sqrt(16/25 + 9/25) = sqrt(25/25) = sqrt(1) = 1.Oh, that's nice, r is 1. So, sin α is just y/r, which is (-3/5)/1 = -3/5. That seems straightforward.Now, moving on to part (2). The expression is:[sin(π/2 - α) / sin(α + π)] - [tan(α - π) / cos(3π - α)]Hmm, this looks a bit complicated, but maybe I can simplify each part using trigonometric identities.Starting with sin(π/2 - α). I remember that sin(π/2 - α) is equal to cos α. So, that simplifies the numerator of the first fraction.Next, sin(α + π). There's an identity for sin(θ + π) which is -sin θ. So, sin(α + π) = -sin α.So, the first fraction becomes cos α / (-sin α) = -cos α / sin α.Now, moving to the second part: tan(α - π). I know that tan has a period of π, so tan(α - π) is the same as tan α. So, that simplifies to tan α.Then, cos(3π - α). Let me think about that. Cosine has an identity for cos(π - θ) = -cos θ, but here it's 3π - α. Let me break it down:cos(3π - α) = cos(2π + π - α) = cos(π - α) because cosine has a period of 2π. And cos(π - α) is -cos α. So, cos(3π - α) = -cos α.So, the second fraction becomes tan α / (-cos α).Putting it all together, the entire expression is:[-cos α / sin α] - [tan α / (-cos α)]Simplify each term:First term: -cos α / sin αSecond term: - [tan α / (-cos α)] = tan α / cos αSo, the expression becomes:(-cos α / sin α) + (tan α / cos α)Now, let's substitute tan α with sin α / cos α:(-cos α / sin α) + (sin α / cos α) / cos αSimplify the second term:(sin α / cos α) / cos α = sin α / (cos α * cos α) = sin α / cos² αSo, now the expression is:(-cos α / sin α) + (sin α / cos² α)To combine these terms, I need a common denominator. Let's see, the denominators are sin α and cos² α. The common denominator would be sin α * cos² α.So, rewrite each term:First term: (-cos α / sin α) = (-cos α * cos² α) / (sin α * cos² α) = (-cos³ α) / (sin α cos² α)Second term: (sin α / cos² α) = (sin α * sin α) / (sin α cos² α) = (sin² α) / (sin α cos² α)So, combining them:(-cos³ α + sin² α) / (sin α cos² α)Hmm, let's see if we can simplify the numerator. Maybe factor something out or use identities.I know that sin² α + cos² α = 1, so sin² α = 1 - cos² α. Let's substitute that into the numerator:- cos³ α + (1 - cos² α) = -cos³ α + 1 - cos² αHmm, that doesn't seem to factor nicely. Maybe I made a mistake earlier.Wait, let me go back a step. Maybe instead of combining them immediately, I can substitute the known values of sin α and cos α.From part (1), we have sin α = -3/5 and cos α = 4/5 (since x is positive, cos α is positive).So, let's substitute these values into the expression:First term: (-cos α / sin α) = (- (4/5) / (-3/5)) = (-4/5) / (-3/5) = (4/5) / (3/5) = 4/3Second term: (tan α / cos α) = ( (-3/4) / (4/5) ) = (-3/4) * (5/4) = (-15)/16Wait, but earlier I had:Expression = (-cos α / sin α) + (tan α / cos α)Which with the values becomes:4/3 + (-15/16) = 4/3 - 15/16To subtract these, find a common denominator. 3 and 16 have 48 as the least common multiple.Convert 4/3 to 64/48 and 15/16 to 45/48.So, 64/48 - 45/48 = 19/48So, the value of the expression is 19/48.Wait, but let me double-check my steps because earlier when I tried to combine the terms algebraically, I ended up with a different expression, but substituting the values gave me 19/48.Let me verify:Starting from:Expression = (-cos α / sin α) + (tan α / cos α)Substitute sin α = -3/5, cos α = 4/5, tan α = sin α / cos α = (-3/5)/(4/5) = -3/4So:First term: (- (4/5) / (-3/5)) = (-4/5) / (-3/5) = (4/5) / (3/5) = 4/3Second term: (-3/4) / (4/5) = (-3/4) * (5/4) = -15/16So, expression = 4/3 - 15/16Convert to common denominator:4/3 = 64/48, 15/16 = 45/48So, 64/48 - 45/48 = 19/48Yes, that seems correct. So, the final answer is 19/48.I think that's it. I didn't need to go through the algebraic manipulation because substituting the known values was straightforward and gave me the answer without complications.

answer：First, I notice that all three numbers inside the cube root—20, 30, and 120—have a common factor of 10. This allows me to factor out (10^3) from each term.Next, I rewrite the expression inside the cube root as:[sqrt[3]{10^3 cdot (2^3 + 3^3 + 12^3)}]I then calculate the sum of the cubes inside the parentheses:[2^3 = 8,quad 3^3 = 27,quad 12^3 = 1728]Adding these together:[8 + 27 + 1728 = 1763]Recognizing that 1763 is equal to (11^3), the expression simplifies to:[10 cdot sqrt[3]{11^3} = 10 cdot 11 = 110]