In how many ways can 10 identical white balls, 10 identical blue balls and 10 identical red balls be distributed in 3 identical boxes?

To determine the number of ways to distribute 10 identical white balls, 10 identical blue balls, and 10 identical red balls into 3 identical boxes, we can use the theory of partitions. We need to partition each set of balls into three parts where the order of parts does not matter.

For each color, we need to find the number of ways to partition 10 balls into up to 3 parts. This is equivalent to finding the number of integer solutions to the equation \( x_1 + x_2 + x_3 = 10 \) with \( x_1, x_2, x_3 \geq 0 \).

The number of non-negative integer solutions to this equation is given by the stars and bars theorem:
\[
\binom{10 + 3 - 1}{3 - 1} = \binom{12}{2}
\]
Calculating this, we get:
\[
\binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66
\]

So, there are 66 ways to distribute the 10 white balls into 3 boxes.

However, since the boxes are identical, we need to consider the partitions of 10 into at most 3 parts, ignoring the order of parts. The number of such partitions is given by the number of ways to partition 10, disregarding order. These partitions are:
\[
\{(10, 0, 0), (9, 1, 0), (8, 2, 0), (8, 1, 1), (7, 3, 0), (7, 2, 1), (6, 4, 0), (6, 3, 1), (6, 2, 2), (5, 5, 0), (5, 4, 1), (5, 3, 2), (4, 4, 2), (4, 3, 3)\}
\]
Counting these, we get 14 distinct partitions.

Thus, the number of ways to partition 10 identical balls of one color into 3 identical boxes is 14. Since we have three colors of balls, the total number of ways to distribute the balls is:
\[
14 \times 14 \times 14 = 14^3 = 2744
\]

Therefore, the total number of ways to distribute 10 identical white balls, 10 identical blue balls, and 10 identical red balls into 3 identical boxes is:
\[
\boxed{2744}
\]

Would you like a detailed breakdown or have any questions?

Here are some related questions to explore further:

1. How would the solution change if the boxes were distinct?
2. What if there were 4 boxes instead of 3?
3. How many ways can you distribute 5 identical balls into 3 identical boxes?
4. How would the problem change if the balls were distinguishable instead of identical?
5. How does the stars and bars method work for solving distribution problems?
6. What are the properties of partitions in number theory?
7. How can generating functions be used to solve similar distribution problems?
8. What is the combinatorial significance of the partition function?

**Tip:** When solving combinatorial problems, identifying whether objects are identical or distinguishable is crucial for determining the appropriate counting method.

Explore how to distribute 10 identical white balls, 10 identical blue balls, and 10 identical red balls into 3 identical boxes using combinatorial analysis. Learn about the Stars and Bars Theorem, partition theory, and calculate the total number of distribution ways. Suitable for advanced high school students.

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