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Answer the question and How would the answer to Question 4 change if all four balls were unique?

The question asks how many ways we can select two balls from four, where three of the balls are of the same color, and one is of a different color.

### Step 1: Breakdown of the situation
- We have 4 balls in total.
- Three balls are of the same color (let's call them \( A, A, A \)).
- One ball is of a different color (call it \( B \)).

### Step 2: Number of ways to select 2 balls
When selecting 2 balls, we have the following possible cases:
1. **Both balls are from the group of three same-colored balls (AAA)**: 
   - We choose 2 balls from the 3 identical ones. The number of ways to choose 2 out of 3 is given by combinations:
   \[
   \text{Ways} = \binom{3}{2} = 3
   \]
2. **One ball from the group of three same-colored balls and the other is the differently colored ball (B)**: 
   - There is only one way to choose 1 ball from the group of three, and one way to choose the different-colored ball. So this gives us:
   \[
   1 \times 1 = 1
   \]

### Step 3: Total number of ways
Summing these two cases:
\[
3 \text{ (from AAA)} + 1 \text{ (from A and B)} = 4
\]

### Final Answer: **4 ways**

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### How the answer changes if all four balls were unique:
If all four balls were unique, we would need to choose 2 balls from 4 unique balls. The number of ways to choose 2 balls out of 4 is:
\[
\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
\]
So, if all four balls were unique, the number of ways to select 2 balls would be **6**.

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Would you like more details or have any questions about this solution?

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Here are 5 related questions to explore:

1. How do we calculate combinations in general?
2. What happens if there are more than two colors of balls?
3. How would the answer change if we could pick 3 balls instead of 2?
4. How would permutations differ from combinations in this context?
5. Can we solve this problem using probability?

### Tip:
When dealing with combinations, remember that the order of selection doesn't matter. If the order matters, you're dealing with permutations instead!

We have four balls, of which three balls are the same color. We select two balls at once. In how many ways can we do that?

Solve how many ways you can select 2 balls from 4, where 3 are the same color and 1 is different. Learn about combinations and how to apply C(n, r) to this problem. Ideal for students in Grades 8-10.

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