Combinations

Step 1: Understand the problem statement.
We need to form a committee of 7 members by selecting from three groups: Teachers (T), Fathers (F), and Students (S). The available members are:
- Teachers (T): 6
- Fathers (F): 5
- Students (S): 4
Step 2: Understand the conditions for committee formation.
Let , , and be the number of teachers, fathers, and students selected, respectively.
The conditions are:
1. Total members: .
2. At least one from each group: , , .
3. Teachers form the majority among them: This means the number of teachers selected must be strictly greater than the number of fathers selected AND strictly greater than the number of students selected. So, and .
Step 3: List all possible combinations of ( ) that satisfy all conditions.
We need to find integer solutions for with , , , and , .
Case 1:
We need .
Also , , and , .
Possible pairs satisfying and , :
- : , , . Here (satisfied).
Number of ways: .
Case 2:
We need .
Also , , and , .
Possible pairs satisfying and , :
- : , , . Here and (satisfied).
Number of ways: .
- : , , . Here and (satisfied).
Number of ways: .
Total for cases: .
Case 3:
We need .
Also , , and , .
Possible pair satisfying and , :
: , , . Here (satisfied).
Number of ways: .
\underline{Case 4: }
We need .
Also , .
There are no pairs such that AND and . (e.g., if , then , which violates ). So, 0 ways for this case.
Step 4: Calculate the total number of ways.
Sum the ways from all valid cases:
Total ways = (Ways for ) + (Ways for ) + (Ways for )
Total ways = .
The final answer is .