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 determine how many different bowls of ice cream the shop can make with three scoops, given that there are seven different flavors. Let's break it down:

### Step 1: Interpret the Problem
Since we're choosing three scoops of ice cream, and the scoops can be of any flavor (repetition of flavors is allowed), this is a **combinations with repetition** problem.

### Step 2: Apply the Formula
The formula for combinations with repetition (also called the stars and bars formula) is:

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

Where:
- \( n \) is the number of different flavors (7 in this case),
- \( k \) is the number of scoops (3 in this case).

### Step 3: Calculate
Using the formula:

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

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

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

### Final Answer:
The ice cream shop can make **84 different bowls of ice cream** with three scoops.

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. How many different bowls of ice cream could be made with two scoops?
2. What if the shop adds two more flavors to their menu? How many three-scoop bowls are possible now?
3. How many different bowls of ice cream could be made if repetition of flavors is not allowed?
4. How would the answer change if each scoop had to be of a different flavor?
5. What if the shop only allowed bowls with an even number of scoops?

**Tip:** The "combinations with repetition" method is useful for problems where order doesn't matter, but repetition of items is allowed.

Learn how to calculate the number of different bowls of ice cream that can be made with three scoops, using combinations with repetition. Understand the formula \( \binom{n + k - 1}{k} \) where \( n \) is the number of flavors and \( k \) is the number of scoops. Suitable for students in Grades 7-9.

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