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?
Number of ways to choose 5 from 9 men
=(9,5)
Number of ways to choose 5 from all 16
=(16,5)
Number of ways to select 5 from 9 men and 7 women with at least one woman
=(16,5)-(9,5)
125
To find the number of ways to select a committee of five people with at least one woman, we can subtract the number of ways to select a committee with no women from the total number of ways to select a committee.
First, let's calculate the total number of ways to select a committee of five from a group of 16 people (7 women + 9 men).
The number of ways to select a committee of five from 16 people is given by the combination formula:
C(16,5) = 16! / (5! * (16-5)!)
Now, let's calculate the number of ways to select a committee of five with no women.
Since we cannot choose any women, our selection must come entirely from the group of nine men. So, we need to calculate the number of ways to select a committee of five from nine men.
C(9,5) = 9! / (5! * (9-5)!)
Finally, to find the number of ways to select a committee with at least one woman, we subtract the number of ways to select a committee with no women from the total number of ways to form a committee:
Total Number of Ways = C(16,5) - C(9,5)