Question

A class contains 9 men and 3 women. Find the number of ways a teacher can select a committee of 4 from the class where there is

Answer 1

at least one woman on the committee.

We can solve this problem by finding the total number of ways to select a committee of 4 from the class, and then subtracting the number of committees with no women.

The total number of ways to select a committee of 4 from 12 people is:

${12 \choose 4} = \frac{12!}{4!8!} = 495$

To count the number of committees with no women, we need to select 4 men from a group of 9:

${9 \choose 4} = \frac{9!}{4!5!} = 126$

So the number of committees with at least one woman is:

$495 - 126 = 369$

Therefore, there are 369 ways for the teacher to select a committee of 4 with at least one woman.