From 6 gentlemen and 4 ladies, a committee of 5 is to be formed. In how many…

Concept: The number of ways to choose r objects from n distinct objects is given by the combination formula, nCr = n! / (r!(n-r)!). When a selection must satisfy an “at least” condition on one sub-group (here, ladies), split it into cases by how many members of that sub-group are included, compute each case with nCr, and add the cases — or, equivalently, subtract the excluded case (zero ladies) from the total unrestricted count.

Application: The committee has 5 members drawn from 6 gentlemen and 4 ladies, with at least 1 lady. List every valid split of ladies and gentlemen and compute each:

1. Case 1 — 1 lady and 4 gentlemen: 4C1 × 6C4 = 4 × 15 = 60

2. Case 2 — 2 ladies and 3 gentlemen: 4C2 × 6C3 = 6 × 20 = 120

3. Case 3 — 3 ladies and 2 gentlemen: 4C3 × 6C2 = 4 × 15 = 60

4. Case 4 — 4 ladies and 1 gentleman: 4C4 × 6C1 = 1 × 6 = 6

5. Sum the cases: 60 + 120 + 60 + 6 = 246

Cross-check (complementary counting): the total ways to choose any 5 of the 10 people is 10C5 = 252; the excluded case of zero ladies (all 5 from the 6 gentlemen) is 6C5 = 6. So at least one lady = 252 − 6 = 246, which matches the case-by-case total above.

Hence, the number of ways to form the committee = 246.