Question

A school dance committee is to consist of 2 freshmen, 3 sophomores, 4 juniors, and 5 seniors. If 7 freshmen, 8 sophomores, 7 juniors, and 7 seniors are eligible to be on the committee, in how many ways can the committee be chosen?

Answer 1

We need to choose 2 freshmen from 7, 3 sophomores from 8, 4 juniors from 7, and 5 seniors from 7.

The number of ways to choose 2 freshmen from 7 is 7C2 = 21.

The number of ways to choose 3 sophomores from 8 is 8C3 = 56.

The number of ways to choose 4 juniors from 7 is 7C4 = 35.

The number of ways to choose 5 seniors from 7 is 7C5 = 21.

To find the total number of ways to choose the committee, we multiply these numbers together:

21 x 56 x 35 x 21 = 2,849,680

Therefore, there are 2,849,680 ways to choose the dance committee.

Answer 2

The problem can be solved using combinations. We need to choose 2 freshmen from 7, 3 sophomores from 8, 4 juniors from 7, and 5 seniors from 7. The number of ways to do this is:

C(7,2) * C(8,3) * C(7,4) * C(7,5) = 21 * 56 * 35 * 21 = 2,756,040.

Therefore, there are 2,756,040 ways to form the dance committee.