how many committee of five people can be chosen from 20 menand 12 women if exactly three men must be on each committee and at least four women on each committee

your wording is gibberish.

How can you have a committee of 5 people "if exactly three men must be on each committee and at least four women on each committee"

Are there two different questions here?

yes, there are two different questions

a) exactly three men must be on each committee
b) at least four women on each committee
thank you

To find the number of committees of five people that can be chosen, we need to consider the given conditions:

1. Exactly three men must be on each committee.
2. At least four women must be on each committee.

Let's break down the problem into smaller steps:

Step 1: Selecting three men from 20 men.
Since exactly three men must be on each committee, we need to choose three men from the given 20 men.
This can be calculated using combination notation: C(20, 3).

C(20, 3) = 20! / (3! * (20 - 3)!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140

Step 2: Selecting two more people from 12 women.
Since we need a total of five people on each committee and three men have already been chosen, we need to select two more people from the given 12 women.
This can be calculated using combination notation: C(12, 2).

C(12, 2) = 12! / (2! * (12 - 2)!) = (12 * 11) / (2 * 1) = 66

Step 3: Combining the selections.
To find the total number of committees, we multiply the number of ways to select three men from 20 and two women from 12.

Total number of committees = C(20, 3) * C(12, 2) = 1140 * 66 = 75,240

Therefore, there are 75,240 committees of five people that can be chosen from 20 men and 12 women, satisfying the given conditions.