A club consists of 2 teachers and 5 students. If a school has 3 teachers and 7 students, in how many ways can clubs be made?

1. Break Down the Problem

We need to form clubs consisting of 2 teachers and 5 students. We will calculate the number of ways to select the teachers and the students independently, and then multiply these results.

2. Relevant Concepts

To determine the number of ways to select groups, we will use the combination formula, represented as:

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

where $n$ is the total number of options, and $r$ is the number of selections to be made.

3. Analysis and Detail

Selecting Teachers

• Total teachers available: 3
• Teachers to select: 2

The number of ways to select 2 teachers from 3 is calculated as follows:

$C(3, 2) = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2 \times 1}{(2 \times 1)(1)} = 3$

Selecting Students

• Total students available: 7
• Students to select: 5

The number of ways to select 5 students from 7 is calculated as follows:

$C(7, 5) = \frac{7!}{5!(7-5)!} = \frac{7!}{5!2!} = \frac{7 \times 6}{2 \times 1} = 21$

4. Verify and Summarize

We have found:

• The number of ways to choose 2 teachers from 3: $3$
• The number of ways to choose 5 students from 7: $21$

Final Answer

The total number of ways to form the club is:

$\text{Total ways} = C(3, 2) \times C(7, 5) = 3 \times 21 = 63$

Thus, the number of ways clubs can be made is 63.