Solving the Selection Dilemma: Choosing a Committee of 5 from 6 Men and 7 Women

[Diane Olive]

Given a club consisting of six men and seven women

a. In how many ways can we select a committee of five persons?

b. In how many ways can we select a committee of five persons with 2 ment and 3 women?

 

[arildno]

This is a very difficult problem.
None here at these forums will be able to solve it, unless YOU POST WHAT YOU HAVE DONE SO FAR!

 

[Avodyne]

Actually, it's not that difficult. But arildno is right that you must show an attempt ...

 

[Dick]

I guess people not showing work is making arildno crabby. Try to avoid this.

 

[HallsofIvy]

Absolutely, you wouldn't like arildno when he is angry!

 

[Diane Olive]

I got it

I went to a study help session last night and was able to get the answer. I would really like to have help in understanding combination, permutation, and other related problems.

 

[Diane Olive]

Here is what we came up with last night:

Given a club consisting of six men and seven women
a. In how many ways can we select a committee of five persons?

This is a fundamental counting principle: 13 X 12 X 11 X 10 X 9 = 154,440

I will find out today if I was right or not.

b. In how many ways can we select a committee of 5 persons with two men and 3 women?

This is 7 choose 3 times 6 choose 2 = 7! divided by 3! X 4! times 6! divided by 2! times 4!
when you simplify I get = 525

 

[arildno]

Here is what we came up with last night:

Given a club consisting of six men and seven women
a. In how many ways can we select a committee of five persons?

This is a fundamental counting principle: 13 X 12 X 11 X 10 X 9 = 154,440

I will find out today if I was right or not.

b. In how many ways can we select a committee of 5 persons with two men and 3 women?

This is 7 choose 3 times 6 choose 2 = 7! divided by 3! X 4! times 6! divided by 2! times 4!
when you simplify I get = 525

Completely correct!
The binomial, \binom{7}{3}\equiv\frac{7!}{3!4!} tells us that
i) There are 7! permutations of 7 women. When these women are divided into two groups, (those 3 to be in the committee and those 4 who don't), then there are 3!*4! different permutations of a particular choice.

Since permutations of a particular choice is irrelevant, we have to divide 7! with 3!4! to get the number of DISTINCT 3-committe groups of women. (i.e, 35)
The number of distinct women groups of 3 is then multiplied with the number of distinct men groups of 2 (15), yielding a total of 525 possible committee choices.

 

[Dick]

Part a) didn't go so well though. Just as arildno explained, the number of ways of picking 3 from 7 is \binom{7}{3}. a) asks you to pick 5 from 13, why isn't the answer \binom{13}{5}?

 

[arildno]

Oops, you are right!

 

[KenKEN Stone]

it's not a difficult problem!