In how many ways can a committee of five people be selected from seven women and nine men if at least one woman must be in the committee?
From 7 women and 9 men = "16 people choose 5" for a committee, there are "16 choose 5" ways, or
(16,5)=C(16,5)=16!/(5!(16-5)!) ways
Out of these, "9 choose 5" ways include no woman, and must be subtracted.
So the number of ways of selecting a committee of 5 from 9 men and 7 women with at least one woman is
"16 choose 5" - "9 choose 5".
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To find the number of ways to select a committee of five people with at least one woman, we can use the concept of counting combinations.
First, let's consider the case when exactly one woman is in the committee.
We have 7 women to choose from and we need to select 1. This can be done in 7 ways.
Next, we need to select 4 more members from the remaining pool of 15 people (9 men and 6 women). This can be done in C(15, 4) ways.
Therefore, the total number of ways to select the committee with exactly one woman is:
7 * C(15, 4)
Now, let's consider the case when exactly two women are in the committee.
We have 7 women to choose from and we need to select 2. This can be done in C(7, 2) ways.
Next, we need to select 3 more members from the remaining pool of 14 people (9 men and 5 women). This can be done in C(14, 3) ways.
Therefore, the total number of ways to select the committee with exactly two women is:
C(7, 2) * C(14, 3)
We can continue this process for when there are exactly three, four, or five women in the committee.
Finally, to get the total number of ways to select a committee with at least one woman, we sum up the results for each case:
Total number of ways = 7 * C(15, 4) + C(7, 2) * C(14, 3) + C(7, 3) * C(13, 2) + C(7, 4) * C(12, 1) + C(7, 5)
Using the formula for combinations, which is C(n, r) = n! / (r! * (n-r)!), you can calculate the exact value.
To solve this problem, we need to consider two scenarios: when there is exactly one woman in the committee and when there are more than one woman in the committee.
1. When there is exactly one woman in the committee:
In this case, we need to select one woman and four people from the remaining individuals, which can be men or women.
- Number of ways to select one woman: 7C1 (selecting 1 woman from 7 women)
- Number of ways to select four people from the remaining individuals: (7+9-1)C(4-1) [since we have selected one woman already, we have 7 women + 9 men - 1 previously selected woman left]
Therefore, the total number of ways when there is exactly one woman is 7C1 * 15C3.
2. When there are more than one woman in the committee:
In this case, we can select two or more women, and the remaining people can be men or women.
- Number of ways to select two women: 7C2 (selecting 2 women from 7 women)
- Number of ways to select three or more people from the remaining individuals: (7+9-2)C(3-2) [since we have selected two women already, we have 7 women + 9 men - 2 previously selected women left]
Therefore, the total number of ways when there are more than one woman is the sum of the ways when there are 2, 3, 4, or 5 women: 7C2 * 14C3 + 7C3 * 14C2 + 7C4 * 14C1 + 7C5 * 14C0.
Now, we can calculate the total number of ways by adding the results from both scenarios:
Total number of ways = 7C1 * 15C3 + 7C2 * 14C3 + 7C3 * 14C2 + 7C4 * 14C1 + 7C5 * 14C0.
Calculating these values will give you the final answer.