Imagine that you are going to buy ice cream. You will buy 2 balls, and can choose between the flavors strawberry, chocolate, vanilla and blueberry.

How many different ice creams can you put together? Write  the different possibilities to show how you think.

Justify why you are sure that you have included all the possibilities.

What if there are more flavors to choose from? How many possibilities are there then?

Let's start by calculating the number of different ice cream combinations you can create with 2 balls, choosing from the flavors strawberry, chocolate, vanilla, and blueberry.

### Step 1: List all possible combinations
Since you are buying 2 balls of ice cream, there are a few different cases to consider:

1. **Both balls have the same flavor.**
2. **The balls have different flavors.**

#### Case 1: Both balls have the same flavor
Here, you can choose the same flavor twice. The possible combinations are:

- Strawberry & Strawberry (SS)
- Chocolate & Chocolate (CC)
- Vanilla & Vanilla (VV)
- Blueberry & Blueberry (BB)

So, there are 4 combinations in this case.

#### Case 2: The balls have different flavors
In this case, you choose two different flavors. Since the order in which you scoop the flavors doesn't matter (e.g., Strawberry & Chocolate is the same as Chocolate & Strawberry), we list each pair only once:

- Strawberry & Chocolate (SC)
- Strawberry & Vanilla (SV)
- Strawberry & Blueberry (SB)
- Chocolate & Vanilla (CV)
- Chocolate & Blueberry (CB)
- Vanilla & Blueberry (VB)

There are 6 different combinations in this case.

### Step 2: Count all possibilities
Adding the combinations from both cases:

- 4 combinations with the same flavor
- 6 combinations with different flavors

Total number of different ice creams = \( 4 + 6 = 10 \).

### List of All Possibilities
Here are all the possible combinations:

1. SS (Strawberry & Strawberry)
2. CC (Chocolate & Chocolate)
3. VV (Vanilla & Vanilla)
4. BB (Blueberry & Blueberry)
5. SC (Strawberry & Chocolate)
6. SV (Strawberry & Vanilla)
7. SB (Strawberry & Blueberry)
8. CV (Chocolate & Vanilla)
9. CB (Chocolate & Blueberry)
10. VB (Vanilla & Blueberry)

### Justification for Including All Possibilities
We have systematically considered all possible scenarios:

- First, we listed the 4 cases where both balls are of the same flavor.
- Then, we listed all 6 unique pairs of different flavors.

By checking all combinations, we ensure that no possibilities are missed. The systematic approach of considering identical and different pairs covers all potential combinations.

### Generalizing to More Flavors
If there are more flavors to choose from, say \( n \) different flavors, the process is similar:

1. **Same Flavor Combinations:** There are \( n \) ways to choose a single flavor for both balls.
2. **Different Flavor Combinations:** To choose 2 different flavors out of \( n \), you use the combination formula:
   
   \[
   \text{Number of combinations} = \binom{n}{2} = \frac{n(n-1)}{2}
   \]

So, the total number of different ice creams you can create is:

\[
\text{Total combinations} = n + \binom{n}{2} = n + \frac{n(n-1)}{2}
\]

For example, if there are 5 flavors:

- **Same Flavor Combinations:** 5
- **Different Flavor Combinations:** \( \binom{5}{2} = \frac{5 \times 4}{2} = 10 \)

Total number of combinations = \( 5 + 10 = 15 \).

Would you like further clarification, or do you have any questions? Here are some related questions to consider:

1. What if you could choose 3 balls of ice cream? How would the combinations change?
2. How does the number of combinations change if the order of the scoops matters?
3. Can you extend the approach to find combinations if you choose from 6 or more flavors?
4. What if you could choose the same flavor more than twice?
5. How would you calculate the combinations if the number of flavors were infinite?

**Tip:** When dealing with combinations, understanding whether the order matters and whether repetitions are allowed is key to choosing the correct method to calculate possibilities.

Learn how to calculate the number of different ice cream combinations when choosing 2 balls from 4 flavors: strawberry, chocolate, vanilla, and blueberry. Understand the cases of choosing the same flavor and different flavors, with step-by-step solutions. Suitable for students in Grades 6-8.

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