Question

From a group of 7 men and 6 women, five persons are to be selected to form a committee so that at least 3 men are there on the committee. In how many ways can it be done?

• A. 564
• B. 546
• C. 746
• D. 756
• E.

Hint:

When selecting persons with specific conditions, break the problem into cases and calculate each case separately, then add the results together.

Answer 1

The Correct Option is D

We need to select a committee of 5 persons, ensuring that at least 3 men are included. We analyze the possible cases. 

Case 1: Select 3 men and 2 women - Ways to choose 3 men from 7: \[ \binom{7}{3} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35. \] - Ways to choose 2 women from 6: \[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15. \] - Total ways for this case: \[ 35 \times 15 = 525. \] 

Case 2: Select 4 men and 1 woman - Ways to choose 4 men from 7: \[ \binom{7}{4} = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35. \] - Ways to choose 1 woman from 6: \[ \binom{6}{1} = 6. \] - Total ways for this case: \[ 35 \times 6 = 210. \] 

Case 3: Select 5 men - Ways to choose 5 men from 7: \[ \binom{7}{5} = \frac{7 \times 6}{2 \times 1} = 21. \] - Total ways for this case: \[ 21. \] 

Total Number of Ways: \[ 525 + 210 + 21 = 756. \] Thus, the total number of ways to form the committee is \( \mathbf{756} \).