In how many ways can a committee of five be chosen from among the 18
teachers and the principal of an elementary school? In what percentage of these committees the
principal will not be included?

Let's solve the two parts of this problem step by step:

### Part 1: Total Number of Ways to Choose the Committee
The total number of people we can choose from is 19 (18 teachers + 1 principal). We need to choose a committee of 5 people from these 19.

The number of ways to choose 5 people from 19 is calculated using the combination formula:

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

where:
- \( n = 19 \) (total number of people),
- \( r = 5 \) (number of people to choose).

Thus, the total number of ways is:

\[
\binom{19}{5} = \frac{19!}{5!(19 - 5)!} = \frac{19!}{5!14!}
\]

Using a calculator:

\[
\binom{19}{5} = \frac{19 \times 18 \times 17 \times 16 \times 15}{5 \times 4 \times 3 \times 2 \times 1} = 11628
\]

So, there are 11,628 ways to choose the committee.

### Part 2: Percentage of Committees Where the Principal Is Not Included
If the principal is not included, we are choosing all 5 people from only the 18 teachers. The number of ways to choose 5 people from 18 teachers is:

\[
\binom{18}{5} = \frac{18!}{5!(18 - 5)!} = \frac{18 \times 17 \times 16 \times 15 \times 14}{5 \times 4 \times 3 \times 2 \times 1} = 8568
\]

Now, to find the percentage of committees where the principal is **not** included:

\[
\text{Percentage} = \left( \frac{\binom{18}{5}}{\binom{19}{5}} \right) \times 100 = \left( \frac{8568}{11628} \right) \times 100 \approx 73.7\%
\]

Thus, approximately **73.7%** of the committees will not include the principal.

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Would you like more details or have any questions? Here are 5 related questions:

1. How many ways can a committee of 4 teachers and the principal be chosen?
2. What if there are 20 teachers instead of 18—how does that affect the result?
3. How many ways can a committee of 3 teachers and 2 principals be chosen?
4. What is the probability that a randomly chosen committee includes exactly 2 teachers?
5. How many committees can be formed where at least one teacher is included?

**Tip:** The combination formula \(\binom{n}{r}\) is useful whenever you are selecting objects without regard to order!

In how many ways can a committee of five be chosen from among the 18 teachers and the principal of an elementary school? In what percentage of these committees the principal will not be included?

Discover how to calculate the number of ways to form a committee of 5 members from 18 teachers and the principal. This combinatorics problem involves using the combination formula to find the total number of selections and determining the percentage of committees where the principal is not included. Suitable for Grades 9-12.

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