Question 1024507: A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 6 freshmen, 7 sophomores, 7 juniors, and 8 seniors are eligible to be on the committee, in how many ways can the committee be chosen?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you need:
2 out of 6 freshmen
3 out of 7 sophomores
4 out of 7 juniors
5 out of 8 seniors
the number of ways you can get 2 out of 6 freshmen is c(6,2) = 15
the number of ways you can get 3 out of 7 sophomores is c(7,3) = 35
the number of ways you can get 4 out of 7 juniors is c(7,4) = 35
the number of ways you can get 5 out of 8 seniors is c(8,5) = 56
the total number of ways you can get 2 freshmen and 3 sophomores and 4 juniors and 5 seniors is therefore 15 * 35 * 35 * 56 = 1,029,000.
c(n,x) is equal to n! / (x! * (n-x)!
for example:
c(8,5) is equal to 8! / (5! * 3!).
this is equal to 8 * 7 * 6 * 5! / (5! * 3!)
the 5! on the top and the 5! on the bottom cancel out and you are left with (8 * 7 * 6) / (3 * 2 * 1).
the 6 on the top and the (3 * 2) on the bottom cancel out and you are left with (8 * 7) / 1 which is equal to 56.
to see how this works, make the problem much smallerr so you can detail out the individual possibilities.
suppose 3 freshmen and 2 sophomores and you want a committee of 2 freshmen and 1 sophomore.
the calculation would be c(3,2) * (c(2,1).
this would be equal to 3 * 2 = 6 possibilities.
assume the freshmen are a,b,c.
assume the sophomores are 1,2.
the possible combinations are:
ab, ac, bc for the freshmen
1, 2 for the sophomores.
the possible teams are therefore ab1, ac1, bc1, ab2, ac2, bc2.
the formula looks good for the smaller set, so you can reasonably apply it to the larger set.
you do have to make sure that the smaller set and the larger set follow the same rules.
if you make it too simple, you might not see the complexities inherent in the larger set.
for practical purposes, this type of analysis does work as long as you are careful that the models work both for the simpler situation and the more complex situaiton.
in this case, it is reasonably certain that it does.
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