Problem: A committee of 3 ladies and 4 gents is to be formed out of 8 ladies and 7 gents. Mrs. X refuses to serve in a committee in which Mr. Y is a member. Find the number of such committees.

Solution:

We can use the principle of inclusion and exclusion to solve this problem. Let us first find the total number of committees that can be formed without any restrictions.

Step 1: Total number of committees:

We need to choose 3 ladies out of 8 ladies and 4 gents out of 7 gents.
Number of ways to choose 3 ladies out of 8 ladies = C(8,3) = 56
Number of ways to choose 4 gents out of 7 gents = C(7,4) = 35
Total number of committees = 56 x 35 = 1960

Step 2: Number of committees with Mr. Y:

We need to choose 3 ladies out of 8 ladies and 3 gents out of 6 gents (excluding Mr. Y).
Number of ways to choose 3 ladies out of 8 ladies = C(8,3) = 56
Number of ways to choose 3 gents out of 6 gents (excluding Mr. Y) = C(6,3) = 20
Total number of committees with Mr. Y = 56 x 20 = 1120

Step 3: Number of committees with Mrs. X and Mr. Y:

We need to choose 2 ladies out of 7 ladies (excluding Mrs. X) and 3 gents out of 6 gents (excluding Mr. Y and Mrs. X).
Number of ways to choose 2 ladies out of 7 ladies (excluding Mrs. X) = C(7,2) = 21
Number of ways to choose 3 gents out of 6 gents (excluding Mr. Y and Mrs. X) = C(6,3) = 20
Total number of committees with Mrs. X and Mr. Y = 21 x 20 = 420

Step 4: Number of committees with Mrs. X:

We need to choose 3 ladies out of 7 ladies (excluding Mrs. X) and 4 gents out of 6 gents (excluding Mr. Y and Mrs. X).
Number of ways to choose 3 ladies out of 7 ladies (excluding Mrs. X) = C(7,3) = 35
Number of ways to choose 4 gents out of 6 gents (excluding Mr. Y and Mrs. X) = C(6,4) = 15
Total number of committees with Mrs. X = 35 x 15 = 525

Step 5: Number of committees without Mr. Y:

Number of committees without Mr. Y = Total number of committees - Number of committees with Mr. Y = 1960 - 1120 = 840

Step 6: Number of committees without Mrs. X and Mr. Y:

Number of committees without Mrs. X and Mr. Y = Number of committees without Mr. Y - Number of committees with Mrs. X and Mr.