Question

The gym has 11 different types of machines in the weight room. Geoff has time to use only 3 of them this afternoon. How many different combinations of machines can Geoff choose from to use? Show your work or explain how you got your answer.

Answer 1

To determine how many different combinations of 3 machines Geoff can choose from 11 different types of machines, we use the concept of combinations. The formula for combinations is given by:

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

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

For this problem:
- \( n = 11 \) (the total number of machines),
- \( r = 3 \) (the number of machines Geoff wants to choose).

Substituting the values into the combinations formula:

\[
C(11, 3) = \frac{11!}{3!(11 - 3)!} = \frac{11!}{3! \cdot 8!}
\]

Next, we simplify \( 11! \) as follows:

\[
11! = 11 \times 10 \times 9 \times 8!
\]

So we can simplify the combinations calculation:

\[
C(11, 3) = \frac{11 \times 10 \times 9 \times 8!}{3! \times 8!}
\]

The \( 8! \) cancels out from the numerator and the denominator:

\[
C(11, 3) = \frac{11 \times 10 \times 9}{3!}
\]

Now, we calculate \( 3! \):

\[
3! = 3 \times 2 \times 1 = 6
\]

Substituting this back into our equation gives:

\[
C(11, 3) = \frac{11 \times 10 \times 9}{6}
\]

Calculating the numerator:

\[
11 \times 10 = 110
\]
\[
110 \times 9 = 990
\]

Now, we divide by 6:

\[
C(11, 3) = \frac{990}{6} = 165
\]

Therefore, the number of different combinations of machines that Geoff can choose from is:

\[
\boxed{165}
\]