Question

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

• A. 1960
• B. 3240
• C. 1540
• D. none of these

Answer 1

The Correct Option is C

To solve the problem, we need to consider two scenarios: when Mr. Y is in the committee and when Mr. Y is not. Given that Mrs. X refuses to serve if Mr. Y is present, these scenarios affect how we count the combinations. We need to calculate the total number of combinations while considering these constraints.

Step 1: Total Committee Combinations Without Restrictions

Without any restrictions, the number of ways to form a committee with 3 ladies out of 8 and 4 gentlemen out of 7 is given by the combination formula:

Combinations = C(n, k) = n! / [k! * (n-k)!]

Calculating separately:

• The number of ways to choose 3 ladies from 8: C(8, 3)=56
• The number of ways to choose 4 gentlemen from 7: C(7, 4)=35

Total combinations of forming the committee without restriction:
Total = C(8, 3) * C(7, 4) = 56 * 35 = 1960

Step 2: Invalid Combinations with Mrs. X and Mr. Y

Next, calculate the number of invalid committees where both Mrs. X and Mr. Y are members:

• Choose 2 ladies from the remaining 7 (as Mrs. X is already chosen): C(7, 2)=21
• Choose 3 gentlemen from the remaining 6 (as Mr. Y is already chosen): C(6, 3)=20

Total invalid combinations with both Mrs. X and Mr. Y:
Total invalid = 21 * 20 = 420

Step 3: Valid Combinations

To find the valid combinations, subtract the invalid combinations from the total possible combinations:

Valid combinations = Total combinations - Invalid combinations = 1960 - 420 = 1540

Thus, the number of ways to form the committee satisfying the given condition is 1540.