how many committee of five people can be chosen from 20 men and 12 women if a) exactly three men must be on each committee
b) at least four women on each committee
Ahh, I see you fixed the problem from your earlier post.
a) 20 men, 12 women, choose 5, with exactly 3 men
= C(20,3) x C(12,2)
= 2053(66)
= 135498
b) at least 4 women
---> 4 women or 5 women
= C(20,1) x C(12,4) + C(20,0) x C(12,5)
= 20(495) + 1(792)
= 10692
check my arithmetic
a) 20 men, 12 women, choose 5, with exactly 3 men
= C(20,3) x C(12,2)
= 2053(66)
= 135498
b) at least 4 women
---> 4 women or 5 women
= C(20,1) x C(12,4) + C(20,0) x C(12,5)
= 20(495) + 1(792)
= 10692
To find the number of committees that meet the given conditions, we can use the combination formula.
a) To form a committee with exactly three men, we first need to select 3 men from the 20 available. We can do this in C(20,3) ways, using the combination formula:
C(n, r) = n! / (r!(n-r)!)
C(20,3) = 20! / (3!(20-3)!) = 20! / (3!17!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140
Now, since there are 12 women available and we need to select 2 more members for the committee, we have to select 2 women from the 12 available. This can be done in C(12,2) ways:
C(12,2) = 12! / (2!(12-2)!) = 12! / (2!10!) = (12 * 11) / (2 * 1) = 66
So, the total number of committees that can be formed with exactly three men and two women is: 1140 * 66 = 75,240.
b) To form a committee with at least four women, we can count the committees with exactly four women, exactly five women, and exactly six women separately, then add them up.
For exactly four women:
We need to select 4 women from the 12 available, which can be done in C(12,4) ways.
C(12,4) = 12! / (4!(12-4)!) = 12! / (4!8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495
For exactly five women:
We need to select 5 women from the 12 available, which can be done in C(12,5) ways.
C(12,5) = 12! / (5!(12-5)!) = 12! / (5!7!) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 792
For exactly six women:
We need to select all 6 women from the 12 available, which can be done in C(12,6) ways.
C(12,6) = 12! / (6!(12-6)!) = 12! / (6!6!) = (12 * 11 * 10 * 9 * 8 * 7) / (6 * 5 * 4 * 3 * 2 * 1) = 924
Now, we can calculate the total number of committees by adding these three possibilities:
495 + 792 + 924 = 2211
So, the total number of committees that can be formed with at least four women is 2,211.