A baking club wants to form an executive committee. There are $15$ people in the baking club, including Mark. In how many ways can the baking club form an executive committee with $2$ people, not including Mark?

Answer 1

To determine the number of ways to form an executive committee with 2 people selected from the baking club's 15 members, excluding Mark, one should follow these steps:

1. First, exclude Mark from the selection pool. That leaves us with \(15 - 1 = 14\) members.
2. We need to select 2 people from these 14 remaining members.

The number of ways to choose 2 people from 14 can be calculated using the combination formula:

\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

Here, \(n = 14\) and \(k = 2\):

\[
\binom{14}{2} = \frac{14!}{2!(14-2)!} = \frac{14!}{2! \cdot 12!}
\]

Next, simplify the factorials:

\[
\binom{14}{2} = \frac{14 \times 13 \times 12!}{2 \times 1 \times 12!}
\]

Cancel the \(12!\) in both the numerator and the denominator:

\[
\binom{14}{2} = \frac{14 \times 13}{2 \times 1} = \frac{182}{2} = 91
\]

Thus, the number of ways to form the executive committee without including Mark is:

\[
\boxed{91}
\]