Find the number of different ways that an instructor can choose 6 students from a class of 22 students for a field trip.


The problem involves choosing 6 students from a class of 22 students. Since the order of selection doesn't matter, we are dealing with a combination.

The formula for combinations is:

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

where:
- \( n \) is the total number of students (22),
- \( r \) is the number of students to be chosen (6).

So, we need to calculate \( C(22, 6) \):

\[
C(22, 6) = \frac{22!}{6!(22 - 6)!} = \frac{22!}{6!16!}
\]

We can simplify this by calculating only the relevant terms from \( 22! \) and \( 16! \):

\[
C(22, 6) = \frac{22 \times 21 \times 20 \times 19 \times 18 \times 17}{6 \times 5 \times 4 \times 3 \times 2 \times 1}
\]

Now, let's compute the value:

\[
C(22, 6) = \frac{22 \times 21 \times 20 \times 19 \times 18 \times 17}{720}
\]

I'll calculate this.The number of different ways the instructor can choose 6 students from a class of 22 students is \( 74,613 \).

Would you like a detailed breakdown of the steps? 

Here are 5 related questions for practice:
1. How many ways can the instructor choose 5 students from a class of 30?
2. How many ways can a team of 4 students be selected from a group of 15?
3. What if the instructor wants to select both a group leader and 5 students from a class of 22? How many ways are there to do this?
4. How many ways can the instructor choose 6 students from 22 if order matters?
5. In how many ways can 7 students be chosen from a class of 25?

**Tip:** In combinations, the order of selection does not matter, but in permutations, the order does!

Find the number of different ways that an instructor can choose 6 students from a class of 22 students for a field trip.

Learn how to calculate the number of ways to choose 6 students from a class of 22 using combinations. This step-by-step guide explains the combination formula and applies it to find the solution. Suitable for Grades 9-12.

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