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

To determine the number of ways to form a 5-member committee from 10 men and 8 women, with the constraint that the committee must include at least 3 women, we analyze the following distinct cases:

1. Case 1: 3 Women and 2 Men

   The number of combinations for selecting 3 women from 8 is calculated using the combination formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).

   \( C(8, 3) = \frac{8!}{3!5!} = 56 \)

   The number of combinations for selecting 2 men from 10 is:

   \( C(10, 2) = \frac{10!}{2!8!} = 45 \)

   The total number of committees for this scenario is the product of these combinations: \( 56 \times 45 = 2520 \)

2. Case 2: 4 Women and 1 Man

   The number of combinations for selecting 4 women from 8 is:

   \( C(8, 4) = \frac{8!}{4!4!} = 70 \)

   The number of combinations for selecting 1 man from 10 is:

   \( C(10, 1) = 10 \)

   The total number of committees for this scenario is: \( 70 \times 10 = 700 \)

3. Case 3: 5 Women and 0 Men

   The number of combinations for selecting 5 women from 8 is:

   \( C(8, 5) = \frac{8!}{5!3!} = 56 \)

   The total number of committees for this scenario is: 56

The total number of committees satisfying the condition is the sum of the committees from all considered scenarios:

• Total committees with at least 3 women = 2520 + 700 + 56 = 3276

Consequently, there are 3276 possible committees that meet the specified criteria.