Question:
From 4 men and 6 ladies a committee of five is to be selected. The number of ways in which the committee can be formed so that men are in majority is
From 4 men and 6 ladies a committee of five is to be selected. The number of ways in which the committee can be formed so that men are in majority is
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Break counting problems into mutually exclusive cases.
Updated On: Apr 30, 2026
- \(68\)
- \(156\)
- \(60\)
- \(72\)
- \(66\)
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The Correct Option is C
Solution and Explanation
Concept:
Majority means number of men \(>\) number of ladies in committee of 5.
Step 1: Possible cases. Total members = 5 Men must be more than women: \[ \text{Possible splits: } (3M,2W), (4M,1W) \]
Step 2: Case 1: 3 men, 2 women \[ \binom{4}{3} \times \binom{6}{2} \] \[ = 4 \times 15 = 60 \]
Step 3: Case 2: 4 men, 1 woman \[ \binom{4}{4} \times \binom{6}{1} \] \[ = 1 \times 6 = 6 \]
Step 4: Total ways. \[ = 60 + 6 = 66 \]
Step 1: Possible cases. Total members = 5 Men must be more than women: \[ \text{Possible splits: } (3M,2W), (4M,1W) \]
Step 2: Case 1: 3 men, 2 women \[ \binom{4}{3} \times \binom{6}{2} \] \[ = 4 \times 15 = 60 \]
Step 3: Case 2: 4 men, 1 woman \[ \binom{4}{4} \times \binom{6}{1} \] \[ = 1 \times 6 = 6 \]
Step 4: Total ways. \[ = 60 + 6 = 66 \]
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