there are 8 women and 6 men in a department. How many ways can a committee of 4 people be selected if there must be at least two women on the committee?
To find the number of ways a committee of 4 people can be selected from a department with 8 women and 6 men, ensuring that there must be at least two women on the committee, we can use the concept of combinations.
First, let's consider the cases in which exactly two women are selected, as this is the minimum requirement:
1. Selecting 2 women and 2 men:
- We can choose 2 women out of 8 in C(8, 2) ways.
- We can choose 2 men out of 6 in C(6, 2) ways.
- The total number of ways to form the committee would be C(8, 2) * C(6, 2).
2. Selecting 3 women and 1 man:
- We can choose 3 women out of 8 in C(8, 3) ways.
- We can choose 1 man out of 6 in C(6, 1) ways.
- The total number of ways to form the committee would be C(8, 3) * C(6, 1).
3. Selecting all 4 women:
- We can choose 4 women out of 8 in C(8, 4) ways.
- Since no men are selected in this case, there will be C(6, 0) ways to select 0 men.
- The total number of ways to form the committee would be C(8, 4) * C(6, 0).
Finally, we add up the results from all three cases to get the total number of ways to form the committee:
Total number of ways = C(8, 2) * C(6, 2) + C(8, 3) * C(6, 1) + C(8, 4) * C(6, 0).
Calculating this expression will give us the answer.