A committee of 4 is to be selected from 5 women and 7 men.In how many ways;a)Can this committee be selected? b)Can this committee be selected if 3 men must be included c)Can the committee be selected if at least 3 women must be included?
5 women
7 men, choose 4
a) number of ways with no restriction
= C(12, 4) = 495
b) must be 3 men and 1 woman
= C(7,3) x C(5,1) = 35 x 5 = 175
c) at least 3 women ---> 3 women or 4 women
= C(5,3) x C(7,1) + C(5,4) x C(7,0)
= 10x7 + 5x1
= 75
To solve these problems, we will use the concept of combinations.
a) To select a committee of 4 from a group of 5 women and 7 men, you can simply calculate the total number of combinations. The formula for combinations is:
nCk = n! / (k!(n-k)!)
where n is the total number of elements and k is the number of elements to be selected.
In this case,
n = 5 women + 7 men = 12
k = 4
So, the number of ways to select the committee is:
12C4 = 12! / (4!(12-4)!) = 12! / (4!8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495.
Therefore, there are 495 ways to select the committee.
b) If 3 men must be included in the committee, we can first select those 3 men from the 7 available, and then fill the remaining spot with either a man or a woman.
First, select 3 men:
7C3 = 7! / (3!(7-3)!) = 7! / (3!4!) = (7 * 6 * 5) / (3 * 2 * 1) = 35.
Next, choose the remaining committee member:
We need to fill only 1 spot with either a man or a woman, so we have a total of 5 women and 4 men left.
Total ways to select 1 person from 9 (5 women + 4 remaining men) = 9C1 = 9.
To get the total number of ways to select the committee, multiply the two results:
35 * 9 = 315.
Therefore, there are 315 ways to select the committee if 3 men must be included.
c) If at least 3 women must be included, we can consider two cases:
1) Selecting exactly 3 women:
Choose 3 women from the 5 available:
5C3 = 5! / (3!(5-3)!) = 5! / (3!2!) = (5 * 4) / (2 * 1) = 10.
Next, choose the remaining committee member:
We can select any person from the 7 men.
7C1 = 7.
Total ways for this case = 10 * 7 = 70.
2) Selecting all 4 women:
Choose all 5 women available:
5C4 = 5.
No men need to be selected in this case.
Total ways for this case = 5.
Finally, add the results from both cases:
70 + 5 = 75.
Therefore, there are 75 ways to select the committee if at least 3 women must be included.