An ice cream shop sells the following flavors: chocolate, vanilla, straw- berry, coffee, butter pecan, mint chocolate chip and raspberry. How many different bowls of ice cream – with three scoops – can they make?

To solve this problem, we need to calculate the number of different combinations of three scoops of ice cream that can be made from the available flavors. Since the order in which the scoops are chosen doesn't matter, and a flavor can be repeated, we use the "combinations with repetition" formula.

The formula for combinations with repetition is given by:

\[
\text{Number of combinations} = \binom{n + r - 1}{r}
\]

Where:
- \( n \) is the number of different flavors available.
- \( r \) is the number of scoops to be selected.

In this case:
- \( n = 7 \) (since there are 7 different flavors: chocolate, vanilla, strawberry, coffee, butter pecan, mint chocolate chip, and raspberry)
- \( r = 3 \) (since we want to select 3 scoops)

So, the calculation is:

\[
\binom{7 + 3 - 1}{3} = \binom{9}{3}
\]

Now, let's calculate \( \binom{9}{3} \):

\[
\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84
\]

Therefore, the number of different bowls of ice cream with three scoops that can be made is **84**.

Would you like more details or have any other questions?

Here are 5 related questions to expand your understanding:
1. How many different two-scoop combinations can be made with these flavors?
2. If the shop introduces an additional flavor, how many three-scoop combinations would there be?
3. What if the scoops had to be all different flavors? How many combinations then?
4. How would the formula change if you could only select each flavor once?
5. What is the probability of randomly selecting a bowl with exactly two scoops of the same flavor?

**Tip:** When dealing with combinations, remember to distinguish between situations where order matters (permutations) and when it doesn't (combinations).

Learn how to calculate the number of different bowls of ice cream that can be made with three scoops from seven available flavors: chocolate, vanilla, strawberry, coffee, butter pecan, mint chocolate chip, and raspberry. Understand the concept of combinations with repetition and apply the formula C(9, 3). Suitable for students in Grades 7-9.

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