a club consists of 25 married couples. determine how many committee of 7 people can be formed from these coules if there are
A. no restrictions
B. exactky 5 men on the committe
c. at least one person has to be a men
d. there must be at least 3 women and 2 men
dgds
To determine the number of committees that can be formed, we need to consider different restrictions and conditions. Let's go through each scenario step by step:
A. No restrictions
In this case, we can select any 7 people out of the 50 individuals (25 married couples) without any limitations. To find the number of possible committees, we can use the combination formula:
C(50, 7) = 50! / (7! * (50 - 7)!)
= 50! / (7! * 43!)
B. Exactly 5 men on the committee
We need to select 5 men from the 25 available and 2 more individuals (can be men or women) from the remaining set. To calculate this, we can use the combination formula again:
C(25, 5) * C(25, 2) = (25! / (5! * (25 - 5)!)) * (25! / (2! * (25 - 2)!))
C. At least one person has to be a man
To determine the number of committees where at least one man is present, we can subtract the number of committees with no men from the total number of committees:
Total number of committees = C(50, 7)
Number of committees with no men = C(25, 7) (selecting 7 women from 25 possible)
Number of committees with at least one man = Total number of committees - Number of committees with no men
D. At least 3 women and 2 men
To calculate the number of committees with at least 3 women and 2 men, we can use a combination again:
[ C(25, 3) * C(25, 2) ] + [ C(25, 4) * C(25, 1) ] + [ C(25, 5) * C(25, 0) ]
This includes selecting 3 women and 2 men, 4 women and 1 man, and 5 women and 0 men respectively.
By substituting the values into the appropriate formula, you can find the numerical values for these scenarios.