First find total number of ways for forming a committee
i.e. 8C3 × 7C4 = 1960
And then find the number of ways in which both Mrs. X and Mr. Y will be there
i.e. 1C1 × 7C2 × 1C1 × 6C3 = 420
Now deduct the number of ways in which both exists from the total number of ways
i.e. 1960 - 420= 1540

Solution:
Given,
Number of ladies = 8
Number of gents = 7
A committee of 3 ladies and 4 gents is to be formed
Mrs X refuses to serve in a committee in which Mr Y is a member.

To find: The number of such committees

Method:

Step 1: Total number of ways to form a committee of 3 ladies from 8 ladies = 8C3
= (8 x 7 x 6)/(3 x 2 x 1) = 56 ways

Step 2: Total number of ways to form a committee of 4 gents from 7 gents = 7C4
= (7 x 6 x 5 x 4)/(4 x 3 x 2 x 1) = 35 ways

Step 3: Total number of ways to form a committee of 3 ladies and 4 gents
= 56 x 35 = 1960 ways

Step 4: Number of ways to form a committee in which Mr Y is a member
= Total number of ways to form a committee of 3 ladies and 3 gents, with Mr Y as one of the gents
= 8C3 x 6C2 = 560 ways

Step 5: Number of ways to form a committee in which Mrs X and Mr Y both are members
= Total number of ways to form a committee of 2 ladies and 2 gents, with Mrs X and Mr Y as one of the gents
= 7C2 x 6C2 = 315 ways

Step 6: Number of ways to form a committee in which Mrs X refuses to serve in a committee in which Mr Y is a member
= Total number of ways to form a committee - Number of ways to form a committee in which Mr Y is a member - Number of ways to form a committee in which Mrs X and Mr Y both are members
= 1960 - 560 - 315 = 1085 ways

Therefore, there are 1085 such committees in which Mrs X refuses to serve in a committee in which Mr Y is a member.