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

To find the number of different combinations of 3 machines that Geoff can choose from 11 different types of machines, we can use the combination formula:

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

where \( n \) is the total number of items to choose from (in this case, 11 machines), \( r \) is the number of items to choose (in this case, 3 machines), and \( ! \) denotes factorial, the product of all positive integers up to that number.

1. Here, \( n = 11 \) and \( r = 3 \).
2. Plugging these values into the combination formula gives:

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

3. We can simplify this calculation:

\[ C(11, 3) = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} \]

4. Calculating the numerator:

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

5. Calculating the denominator:

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

6. Dividing the results:

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

Thus, the total number of different combinations of machines that Geoff can choose from is \( \boxed{165} \).