Question 933503
Ten women and ten men are on the faculty in the mathematics department 
at a school.
 
- How many ways are there to select a committee of five members of the
department if at least one woman must be on the committee?
<pre>
1. We find the number of ways to select any 5 people, regardless of gender.
2. We find the number of ways to select 5 men.
3. We subtract the result of 2 from the result of 1.

1. That's C(20,5)
2. That's C(10,5)
3. That's C(20,5)-C(10,5) = 15504-252 = 15252 
</pre>
- How many ways are there to select a committee of five members of the
department if at least one woman and at least one man must be on the committee?
<pre> 
1. We find the number of ways to select any 5 people, regardless of gender.
2. We find the number of ways to select 5 men.
3. We find the number of ways to select 5 women.
4. We subtract the result of 2 and the result of 3 from the result of 1.

1. That's C(20,5)
2. That's C(10,5)
3. That's C(10,5)
4. That's C(20,5)-C(10,5)-C(10,5) = 15504-252-252 = 15000.

Edwin</pre>