Find the number of ways of selecting 10 balls out of an unlimited number of identical white, red, and blue balls.

Explanation

To solve the problem of selecting 10 balls from an unlimited number of identical white, red, and blue balls, we can use the concept of combinations with repetition.

### Step-by-step Solution:

1. **Define Variables**:
Let \( X \) be the number of white balls, \( Y \) be the number of red balls, and \( Z \) be the number of blue balls. We need to find the number of non-negative integer solutions to the equation:
\[ X + Y + Z = 10 \]

**Hint**: Identify the variables representing the different types of balls.

2. **Use the Stars and Bars Theorem**:
The problem of distributing \( n \) identical items (balls) into \( r \) distinct groups (colors of balls) can be solved using the stars and bars theorem. The formula to find the number of ways to distribute \( n \) identical items into \( r \) distinct groups is given by:
\[ \binom{n + r - 1}{r - 1} \]

**Hint**: Recall the stars and bars theorem for distributing identical items into distinct groups.

3. **Substitute Values**:
In our case, \( n = 10 \) (the total number of balls) and \( r = 3 \) (the three colors: white, red, and blue). Plugging these values into the formula gives:
\[ \binom{10 + 3 - 1}{3 - 1} = \binom{12}{2} \]

**Hint**: Make sure to adjust \( n \) and \( r \) correctly in the formula.

4. **Calculate the Combination**:
Now, we need to calculate \( \binom{12}{2} \):
\[ \binom{12}{2} = \frac{12 \times 11}{2 \times 1} = \frac{132}{2} = 66 \]

**Hint**: Remember to simplify the combination formula step by step.

5. **Conclusion**:
Therefore, the number of ways to select 10 balls from an unlimited number of identical white, red, and blue balls is:
\[ \boxed{66} \]

**Hint**: Always box your final answer to highlight it clearly.