Question

The number of ways in which a committee of 3 ladies and 4 gentlemen can be appointed out of 8 ladies and 7 gentlemen, if Mrs. X refuses to serve in a committee of which Mr. Y is a member, is

• A. 1,540
• B. 1,960
• C. 3,240
• D. None of these

Hint:

For "A will not serve with B" constraints, use "Total \(-\) Together" counting: compute all possibilities, then subtract the cases where the incompatible pair appears together.

Answer 1

The Correct Option is A

Step 1: Count without restriction.
Choose ladies: \(\binom{8}{3}=56\). Choose gentlemen: \(\binom{7}{4}=35\).
Total committees (no restriction) \(= 56 \times 35 = 1960\). Step 2: Subtract forbidden committees (both Mrs. X and Mr. Y included).
If both are included, then:
\(\bullet\) Ladies: Mrs. X is fixed; choose remaining \(2\) from the other \(7\) ladies \(\Rightarrow \binom{7}{2}=21\).
\(\bullet\) Gentlemen: Mr. Y is fixed; choose remaining \(3\) from the other \(6\) gentlemen \(\Rightarrow \binom{6}{3}=20\).
Forbidden count \(= 21 \times 20 = 420\). Step 3: Apply restriction.
Valid committees \(= 1960 - 420 = \boxed{1540}\).