Sub-committee Formation Calculation

The problem asks for the number of ways to choose a sub-committee of 4 members from a larger group of 9 members. Since the order in which members are selected does not matter, this is a combination problem.

Combination Formula

The number of combinations of selecting k items from a set of n items is calculated using the binomial coefficient formula:

$ C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!} $

Applying the Formula

In this case, we have:

• Total number of members, n = 9
• Number of members to be selected for the sub-committee, k = 4

Substitute these values into the formula:

$ C(9, 4) = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!} $

Step-by-Step Calculation

1. Expand the factorials:

   $ C(9, 4) = \frac{9 \times 8 \times 7 \times 6 \times 5!}{ (4 \times 3 \times 2 \times 1) \times 5! } $

2. Cancel out the common 5! term:

   $ C(9, 4) = \frac{9 \times 8 \times 7 \times 6}{ 4 \times 3 \times 2 \times 1 } $

3. Simplify the denominator: $4 \times 3 \times 2 \times 1 = 24$.

   $ C(9, 4) = \frac{9 \times 8 \times 7 \times 6}{ 24 } $

4. Perform the multiplication in the numerator: $9 \times 8 \times 7 \times 6 = 3024$.

   $ C(9, 4) = \frac{ 3024 }{ 24 } $

5. Divide to find the final result:

   $ C(9, 4) = 126 $

Conclusion

There are 126 different ways to form a sub-committee of 4 members from a club of 9 members.