Question

Find the number of ways to form a 7-member committee consisting of 8 parents, 5 teachers, and the principal.

Answer 1

To form a 7-member committee from a group consisting of 8 parents, 5 teachers, and 1 principal, we need to consider various combinations of the members. The total number of individuals available is:

\[
\text{Total members} = 8 \text{ (parents)} + 5 \text{ (teachers)} + 1 \text{ (principal)} = 14
\]

We need to form a committee of 7 members which can include any combination of parents, teachers, and the principal. The principal can either be included in the committee or not, which leads us to consider two cases:

### Case 1: The Principal is included in the committee

If the principal is included, we need to select 6 more members from the remaining 13 individuals (8 parents + 5 teachers):

\[
\text{Number of ways} = \binom{13}{6}
\]

Calculating \(\binom{13}{6}\):

\[
\binom{13}{6} = \frac{13 \times 12 \times 11 \times 10 \times 9 \times 8}{6 \times 5 \times 4 \times 3 \times 2 \times 1} = \frac{1235520}{720} = 1716
\]

### Case 2: The Principal is not included in the committee

In this case, we need to select all 7 members from the 13 individuals (8 parents + 5 teachers):

\[
\text{Number of ways} = \binom{13}{7}
\]

Calculating \(\binom{13}{7}\):

Since \(\binom{13}{7} = \binom{13}{6}\) (because \(\binom{n}{k} = \binom{n}{n-k}\)), we have:

\[
\binom{13}{7} = 1716
\]

### Total number of ways to form the committee

To find the total number of ways to form the committee, we need to sum the number of ways from both cases:

\[
\text{Total} = \binom{13}{6} + \binom{13}{7} = 1716 + 1716 = 3432
\]

Thus, the total number of ways to form a 7-member committee is:

\[
\boxed{3432}
\]