Formation of the Committee

To form a committee, we need to select 3 ladies and 4 gents from a group of 8 ladies and 7 gents. However, there is a condition that Mrs. refuses to serve in a committee if Mr. Y is a member.

Step 1: Selecting the ladies
Since we need to select 3 ladies, we can use the combination formula. The number of ways to select 3 ladies from 8 is given by:
C(8, 3) = 8! / (3! * (8-3)!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56

Step 2: Selecting the gents
Similarly, we need to select 4 gents from a group of 7. The number of ways to do this is given by:
C(7, 4) = 7! / (4! * (7-4)!)
= (7 * 6 * 5) / (3 * 2 * 1)
= 35

Step 3: Considering the condition
Now, we need to consider the condition that Mrs. refuses to serve in a committee if Mr. Y is a member. Let's assume Mr. Y is one of the selected gents. In this case, we have:
Number of ways to select Mr. Y = 1
Number of ways to select the remaining 3 gents = C(6, 3) = 6! / (3! * (6-3)!) = 20

Step 4: Subtracting the restricted cases
To find the number of committees that satisfy the given condition, we need to subtract the number of restricted cases from the total number of possible committees.

Total number of possible committees = Number of ways to select ladies * Number of ways to select gents
= 56 * 35
= 1960

Number of committees with Mr. Y as a member = Number of ways to select ladies * Number of ways to select gents (including Mr. Y)
= 56 * 20
= 1120

Number of committees without Mr. Y = Total number of possible committees - Number of committees with Mr. Y as a member
= 1960 - 1120
= 840

Conclusion
Therefore, the number of committees that can be formed, satisfying the given condition, is 840.