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.

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

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

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

Let's break down the problem step by step.

### Part (a): Number of Different Ice Cream Combinations
You want to buy 2 scoops of ice cream, and you can choose between four different flavors: strawberry (S), chocolate (C), vanilla (V), and blueberry (B). 

Since the order of the scoops doesn’t matter (i.e., a combination of strawberry and chocolate is the same as chocolate and strawberry), you can calculate the number of different combinations using the combination formula.

For 2 scoops from 4 flavors:

\[
\text{Number of combinations} = \binom{n}{k} = \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
\]

### Listing the Combinations:
Now, let's list out all the possible combinations:
1. **Strawberry and Chocolate (S, C)**
2. **Strawberry and Vanilla (S, V)**
3. **Strawberry and Blueberry (S, B)**
4. **Chocolate and Vanilla (C, V)**
5. **Chocolate and Blueberry (C, B)**
6. **Vanilla and Blueberry (V, B)**

So, there are 6 different combinations.

### Part (b): Justification
To ensure all possibilities are included, let's look at the logic:

- We systematically paired each flavor with every other flavor. For example, starting with strawberry, we paired it with chocolate, vanilla, and blueberry. Then, we moved to chocolate and paired it with vanilla and blueberry (excluding strawberry since it's already paired with chocolate in another combination). 
- The combinations are distinct because the order doesn't matter (i.e., S and C is the same as C and S).

This systematic approach ensures that no combinations are missed or duplicated.

### Part (c): More Flavors
Now, let's consider what happens if there are more flavors available.

If you have \( n \) different flavors and you want to choose 2 scoops, the number of combinations is given by the combination formula:

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

This formula calculates how many unique pairs of flavors you can select from \( n \) options.

For example:
- If there are 5 flavors, the number of combinations is \( \binom{5}{2} = 10 \).
- If there are 6 flavors, the number of combinations is \( \binom{6}{2} = 15 \).

### Summary:
- **(a)** There are 6 different combinations of ice creams with 2 scoops from 4 flavors.
- **(b)** The systematic pairing of each flavor with others ensures all combinations are included.
- **(c)** The number of combinations for \( n \) flavors is \( \frac{n(n-1)}{2} \).

Would you like more details or have any further questions?

### Relative Questions:
1. What if you could choose 3 scoops instead of 2? How many combinations would there be?
2. How does the problem change if the order of scoops matters?
3. What if you could choose the same flavor for both scoops? How many combinations would there be then?
4. How would the number of combinations change if there were only 3 flavors to choose from?
5. What happens if you have an option to add a topping to your ice cream? How does that affect the combinations?

### Tip:
Whenever you're unsure if you've considered all combinations, try listing them out systematically to catch any that might have been missed!

Learn how to calculate the number of different ice cream combinations when choosing 2 scoops from 4 flavors (strawberry, chocolate, vanilla, and blueberry). Includes step-by-step explanation, listing all combinations, and exploring scenarios with more flavors.

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