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

Answer 1

The Correct Option is D

We need to select 5 persons with at least 3 men. We can do this in the following ways:

1. Select 3 men and 2 women:
   • Number of ways to select 3 men from 7:

\( ^7C_3 \) = \( \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \) 

• Number of ways to select 2 women from 6:

\( ^6C_2 \) = \( \frac{6 \times 5}{2 \times 1} = 15 \)

• Total ways for this case: \( 35 \times 15 = \mathbf{525} \)

1. Select 4 men and 1 woman:
   • Number of ways to select 4 men from 7:

\( ^7C_4 \) = \( \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} = 35 \)

• Number of ways to select 1 woman from 6:

\( ^6C_1 \) = 6

• Total ways for this case: \( 35 \times 6 = \mathbf{210} \)

1. Select 5 men:
   • Number of ways to select 5 men from 7:

\( ^7C_5 \) = \( \frac{7 \times 6}{2 \times 1} = 21 \)

• Total ways for this case: 21

Thus, the total number of ways is:

\( 525 + 210 + 21 = 756 \)

Thus, the total number of ways to form the committee is 756.