Question

From a group of 10 men and 8 women, a committee of 5 is to be formed such that it contains at least 3 women. How many such committees are possible?

• A. 3276
• B. 3500
• C. 3657
• D. 4012

Hint:

To find the number of committees with at least \(k\) women, sum over cases from \(k\) to the committee size, calculating combinations accordingly.

Answer 1

The Correct Option is A

Total men = 10, total women = 8, committee size = 5, with at least 3 women. Possible cases: - 3 women and 2 men - 4 women and 1 man - 5 women and 0 men Number of committees with 3 women and 2 men: \[ \binom{8}{3} \times \binom{10}{2} = 56 \times 45 = 2520 \] Number of committees with 4 women and 1 man: \[ \binom{8}{4} \times \binom{10}{1} = 70 \times 10 = 700 \] Number of committees with 5 women and 0 men: \[ \binom{8}{5} \times \binom{10}{0} = 56 \times 1 = 56 \] Total number of committees: \[ 2520 + 700 + 56 = 3276 \]