Question

From a group of 5 men and 7 women, five persons are to be selected to form a committee so that at least 3 men are there in the committee. The number of ways this can be done is

• A. 181
• B. 274
• C. 246
• D. None of these

Hint:

In selection problems, break the problem down into cases based on conditions like "at least" and calculate the number of ways for each case.

Answer 1

The Correct Option is C

Step 1: Identify possible cases.
We need to select 5 persons with at least 3 men in the committee. We can have the following cases: - Case 1: 3 men and 2 women - Case 2: 4 men and 1 woman - Case 3: 5 men and 0 women
Step 2: Calculate number of ways for each case.
- Case 1: Number of ways to select 3 men from 5 and 2 women from 7: \[ \binom{5}{3} \times \binom{7}{2} = 10 \times 21 = 210 \] - Case 2: Number of ways to select 4 men from 5 and 1 woman from 7: \[ \binom{5}{4} \times \binom{7}{1} = 5 \times 7 = 35 \] - Case 3: Number of ways to select 5 men from 5: \[ \binom{5}{5} = 1 \]
Step 3: Total number of ways.
Total = 210 + 35 + 1 = 246.
Step 4: Conclusion.
Thus, the number of ways to form the committee is 246, which corresponds to option (C).