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?

Answer 1

To determine the number of ways to form a committee with 2 people from a group of 15, we will use the concept of combinations. A combination is used because the order in which the committee members are chosen does not matter.

The formula to calculate combinations is given by:
\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]
where \( n \) is the total number of people to choose from, and \( r \) is the number of people to be chosen.

In this problem, \( n = 15 \) and \( r = 2 \). Plugging these values into the formula, we get:
\[
\binom{15}{2} = \frac{15!}{2!(15-2)!} = \frac{15!}{2! \cdot 13!}
\]

Since we have \( 15! = 15 \times 14 \times 13! \), we can simplify the factorials:
\[
\binom{15}{2} = \frac{15 \times 14 \times 13!}{2! \times 13!} = \frac{15 \times 14}{2!}
\]

We know that \( 2! = 2 \times 1 = 2 \), so we continue simplifying:
\[
\binom{15}{2} = \frac{15 \times 14}{2} = \frac{210}{2} = 105
\]

Thus, the number of ways to form an executive committee with 2 people from the baking club is:
\[
\boxed{105}
\]