Question

From a group of 7 men and 5 women, a 4-person committee is to be chosen at random. What is the probability that it will have 2 men and 2 women?

Answer 1

`0.bar42`

Explanation:

We can look at how we can pick 2 men and 2 women and compare that to the number of ways we can pick the committee without the gender limitation.

Picking a committee is a combination question - we don't care about the order in which the members are picked.

`C_(n,k)=((n),(k))=(n!)/((k!)(n-k)!)` with `n="population", k="picks"`

2 men

`C_(7,2)=((7),(2))=(7!)/((2!)(5!))=21`

2 women

`C_(5,2)=((5),(2))=(5!)/((2!)(3!))=10`

Total ways for 2 men and 2 women

`21xx10=210`

Committee picked from all 12 people

`C_(12,4)=((12),(4))=(12!)/((4!)(8!))=495`

Probability

`210/495=0.bar42`