Question

A school club consists of 10 members. The club needs to form a committee of 4 members to organize an event. How many different ways can the committee be formed?

Is this a combination or a permutation? (2 points)
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How many different ways can a committee be formed? (5 points)

Answer 1

To determine how many different ways a committee of 4 members can be formed from a school club consisting of 10 members, we need to use combinations because the order in which the members are selected does not matter.

The number of ways to choose 4 members from a group of 10 is given by the combination formula:

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

where:
- \( n \) is the total number of members (10 in this case),
- \( r \) is the number of members to choose (4 for the committee),
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

Plugging the values into the formula, we get:

\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!}
\]

Calculating the factorials:

- \( 10! = 10 \times 9 \times 8 \times 7 \times 6! \) (we can stop here since \( 6! \) will cancel out)
- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 6! \) cancels out so we only need \( 10 \times 9 \times 8 \times 7 \).

Now we can simplify:

\[
\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = \frac{5040}{24} = 210
\]

So, the number of different ways to form the committee is **210**.

### Summary:
1. This is a **combination** (order does not matter).
2. The number of different ways to form the committee is **210**.