Question 1170761: A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 7 freshmen, 9 sophomores, 9 juniors, and 9 seniors are eligible to be on the committee, in how many ways can the committee be chosen?
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Answer by math_tutor2020(3817)   (Show Source):
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We have 7 freshmen total, and we want to pick 2 of them. There are 7*6 = 42 ways to do this if order mattered; however it doesn't. So we divide by 2 to get 42/2 = 21. We could use the nCr combination formula with n = 7 and r = 2.
There are 9 sophomores and 3 spots for them. There are 9*8*7 = 504 permutations and 504/(3!) = 504/(3*2*1) = 504/6 = 84 combinations. We divide by 3! = 6 because there are 6 ways to arrange any group of three people.
There are 9 juniors and 4 slots for them. Let's use the nCr combination formula with n = 9 and r = 4 and we get
nCr = (n!)/(r!*(n-r)!)
9C4 = (9!)/(4!*(9-4)!)
9C4 = (9!)/(4!*5!)
9C4 = (9*8*7*6*5!)/(4!*5!)
9C4 = (9*8*7*6)/(4!) a pair of "5!" terms cancel
9C4 = (9*8*7*6)/(4*3*2*1)
9C4 = 3024/24
9C4 = 126
There are 126 ways to pick the four juniors from a total of nine.
Finally, there are 9 seniors and 5 seats for them on the committee. Plug in n = 9 and r = 5
nCr = (n!)/(r!*(n-r)!)
9C5 = (9!)/(5!*(9-4)!)
9C5 = (9!)/(5!*5!)
9C5 = (9*8*7*6*5!)/(5!*4!)
9C5 = (9*8*7*6)/(4!)
9C5 = (9*8*7*6)/(4*3*2*1)
9C5 = 3024/24
9C5 = 126
There are 126 ways to pick the five seniors from a total of nine.
We get the same result because of the symmetry found in the combination formula.
You could use Pascal's triangle to get this value, or any other value found earlier. In this case, look in the row that starts with "1,9,...". Start the counter at 0 and count up until you get to r = 5. Each time you increase the counter, you move to the right 1 spot. Following this process should lead you to 126.
So we have found
21 ways to pick the freshmen
84 ways to pick the sophomores
126 ways to pick the juniors
126 ways to pick the seniors
Multiply those values out: 21*84*126*126 = 28,005,264
Answer: 28,005,264 different committees.
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