A committee of 5 people is selected from a group of 7 men and 9 women. In how many ways can the committee be selected so it contains at least 1 woman?
What do you think?
In how many ways can a committee of 4 be selected from a group of 7 female and 5 male students so that there are exactly 2 female students?
contains at least one woman:
16C5-7C5
where C means choose. Enter on a calculator to find the value.
To find the number of ways to form a committee with at least 1 woman, we need to consider two scenarios: selecting exactly one woman and selecting more than one woman.
1. Selecting exactly one woman:
We have 9 women to choose from and we want to select only one. Therefore, there are 9 ways to choose exactly one woman. After choosing a woman, we need to select the remaining 4 members from the total available pool of 16 people (7 men + 9 women - 1 woman already chosen). This can be done in C(16, 4) ways, which is the number of combinations of selecting 4 members from 16. Therefore, the total number of ways to form a committee by selecting exactly one woman is 9 * C(16, 4).
2. Selecting more than one woman:
We can select 2, 3, 4, or all 5 women for the committee. Let's consider each scenario:
- Selecting 2 women:
We have 9 women to choose from, so there are C(9, 2) ways to select 2 women. After selecting 2 women, we need to select the remaining 3 members from the remaining pool of 14 people (7 men + 9 women - 2 women already chosen). This can be done in C(14, 3) ways. Therefore, the number of ways to form a committee by selecting exactly 2 women is C(9, 2) * C(14, 3).
- Selecting 3 women:
We have C(9, 3) ways to select 3 women and C(14, 2) ways to select the remaining 2 members. Therefore, the number of ways to form a committee by selecting exactly 3 women is C(9, 3) * C(14, 2).
- Selecting 4 women:
We have C(9, 4) ways to select 4 women and C(14, 1) ways to select the remaining 1 member. Therefore, the number of ways to form a committee by selecting exactly 4 women is C(9, 4) * C(14, 1).
- Selecting all 5 women:
We have C(9, 5) ways to select all 5 women. In this case, there are no remaining members to select since all the spots in the committee have been filled.
To find the total number of ways to form a committee with at least 1 woman, we need to sum up the number of ways from both scenarios:
Total ways = 9 * C(16, 4) + C(9, 2) * C(14, 3) + C(9, 3) * C(14, 2) + C(9, 4) * C(14, 1) + C(9, 5)
Now you can substitute the values and calculate the total number of ways.