Permutations & Combinations
Permutation and Combination
star_batch_jee_advanced_2025
Grade 11
Question:
From 4 men and 6 ladies a committee of 5 is to be selected. The number of ways in which the committee can be formed so that men are in majority, is:
Step-by-Step Solution
Key Concept: Majority means strictly more than half, so with 5 people we need at least 3 men, requiring us to consider exactly 3 or exactly 4 men cases.
For men to be in majority in a committee of 5, we need more than 2.5 men, so at least 3 men. The possible cases are: (1) 3 men and 2 ladies: $\binom{4}{3} \times \binom{6}{2} = 4 \times 15 = 60$, and (2) 4 men and 1 lady: $\binom{4}{4} \times \binom{6}{1} = 1 \times 6 = 6$. Total ways = $60 + 6 = 66$.
Correct Answer: 1