Problem Statement:
A committee of 9 persons is to be made from 11 boys and 7 girls so as to include 5 boys and 4 girls. In how many ways can it be done if 3 particular boys always join the same party?

Solution:

Step 1: Determine the number of ways to select 3 particular boys who always join the same party.

- Since 3 particular boys always join the same party, we only need to select 2 more boys from the remaining 8 boys.
- Number of ways to select 2 boys from 8 boys = 8C2 = 28 ways.

Step 2: Determine the number of ways to select 5 boys from the remaining 8 boys.

- Since 3 particular boys are already selected, we only need to select 2 more boys from the remaining 8 boys.
- Number of ways to select 2 boys from 8 boys = 8C2 = 28 ways.

Step 3: Determine the number of ways to select 4 girls from 7 girls.

- Number of ways to select 4 girls from 7 girls = 7C4 = 35 ways.

Step 4: Determine the total number of ways to form the committee.

- Total number of ways to form the committee = (number of ways to select 3 particular boys who always join the same party) x (number of ways to select 5 boys from the remaining 8 boys) x (number of ways to select 4 girls from 7 girls)
- Total number of ways to form the committee = 28 x 28 x 35 = 27440 ways.

Therefore, there are 27440 ways to form a committee of 9 persons including 3 particular boys who always join the same party, 5 boys, and 4 girls from a group of 11 boys and 7 girls.