In how many ways can 30 balls of 10 different colors, 3 identical ball of each color, be randomly distributed in 10 similar boxes?

To determine the number of ways to distribute 30 balls of 10 different colors (with each color having 3 identical balls) into 10 identical boxes, we need to consider both the constraints of identical balls of the same color and the identical boxes.

### Step-by-Step Solution:

1. **Total Number of Balls**:
   We have a total of 30 balls (since \(10 \times 3 = 30\)).

2. **

Explore how to distribute 30 balls of 10 different colors, with each color having 3 identical balls, into 10 identical boxes. This combinatorial problem involves calculating the number of ways to arrange identical items in distinct groups. Suitable for advanced high school students.

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