Question

If a committee of 10 members is to be formed from 8 men and 6 women, then the number of different possible committees in which the men are in majority is:

• A. 931
• B. 175
• C. 48
• D. 595

Hint:

To form a committee with restrictions like the majority of one group, break down the problem into cases based on the possible numbers and use combinations to count the possibilities for each case.

Answer 1

The Correct Option is D

We are given 8 men and 6 women, and we need to form a committee of 10 members in such a way that the men are in the majority. Since the committee must consist of 10 members and the men must be in the majority, the committee must have at least 6 men and at most 7 men. 
Step 1: We will consider two cases based on the number of men in the committee: 
Case 1: 6 men and 4 women. 
The number of ways to select 6 men from 8 is given by: \[ \binom{8}{6} = \frac{8 \times 7}{2 \times 1} = 28. \] The number of ways to select 4 women from 6 is given by: \[ \binom{6}{4} = \frac{6 \times 5}{2 \times 1} = 15. \] Thus, the total number of committees in this case is: \[ 28 \times 15 = 420. \] Case 2: 7 men and 3 women. 
The number of ways to select 7 men from 8 is given by: \[ \binom{8}{7} = 8. \] The number of ways to select 3 women from 6 is given by: \[ \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20. \] Thus, the total number of committees in this case is: \[ 8 \times 20 = 160. \] 
Step 2: The total number of ways to form the committee with the men in majority is the sum of the results from the two cases: \[ 420 + 160 = 595. \] Conclusion: 
Thus, the number of different possible committees in which the men are in the majority is \( 595 \).