Comination Problem

Question

 

7 freshmen, 7 sophomores, 7 juniors, and 7 seniors are eligible to be on a committee.

 

In how many ways can a dance committee of 17 students be chosen if it is to consist of at least 1 freshmen, 1 sophomore, 2 juniors, 2 seniors, and the rest can be of any grade?

Answer 1

Quite a few!

There are 7 ways to choose the designated freshman

   7 ways to choose the designated sophomore

   7 x 6 / 2 = 21 ways to choose the two designated juniors

   also 21 ways to choose the two designated seniors.

 

  Having selected these 6 students, the remaining pool has 28 - 6 = 22 people.

   From this pool 17 - 6 = 11 must be chosen.    The number of ways of doing this is   22! /( 11! 11!)

 

So the final number is 7 x 7 x 21 x 21 x  22!/( 11!)²

 

This is a huge number.