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

To determine the number of ways to distribute 30 balls of 10 different colors, with 3 identical balls for each color, into 10 similar boxes, we can use combinatorial methods. Let's analyze the problem step by step.

1. **Understanding the problem:**
   - We have 30 balls in total, with 10 different colors and 3 balls of each color.
   - We need to distribute these 30 balls into 10 identical boxes.

2. **Applying combinatorial principles:**
   - Because the boxes are identical, the problem simplifies. Each color has three balls, which need to be distributed among the boxes.

3. **Distribution of each color:**
   - For each color, we need to distribute 3 identical balls into 10 identical boxes.
   - The number of ways to distribute \( k \) identical balls into \( n \) identical boxes is given by the formula for combinations with repetition: \(\binom{n+k-1}{k}\).
   - In our case, for one color, it would be \(\binom{10+3-1}{3} = \binom{12}{3}\).

4. **Calculating \(\binom{12}{3}\):**
   \[
   \binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220
   \]

5. **Considering all colors:**
   - Since there are 10 different colors, and the distribution of each color is independent of the others, the total number of ways to distribute all the balls is \( (\binom{12}{3})^{10} \).

6. **Final calculation:**
   \[
   (\binom{12}{3})^{10} = 220^{10}
   \]

Therefore, the total number of ways to distribute the 30 balls of 10 different colors into 10 identical boxes is \( 220^{10} \).

Would you like a detailed breakdown of any part of this solution? Here are some related questions:

1. What is the general formula for distributing \( k \) identical items into \( n \) identical boxes?
2. How do you derive the combination formula used in this context?
3. How would the solution change if the boxes were distinguishable?
4. What is the significance of using combinations with repetition in this problem?
5. How can the problem be approached using generating functions?
6. How would the solution change if the balls were not identical?
7. What if there were restrictions on the number of balls each box could contain?
8. Can you explain the difference between distributing identical versus distinct items into identical boxes?

**Tip:** When solving combinatorial problems, it is crucial to identify whether the objects and containers are identical or distinguishable, as it significantly affects the counting method.

Explore how to distribute 30 balls of 10 different colors, each with 3 identical pieces, into 10 identical boxes using combinatorial methods. Understand the combinations with repetition formula and calculate the total number of distributions. Suitable for high school students learning combinatorial mathematics.

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