- #1

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- Summary:
- How to count the number of ways to distribute N balls into M piles where the balls are indistinguishable?

If there are N

$$\binom{M+N-1}{N}$$

However, if the boxes are themselves

**distinguishable**boxes and M**indistinguishable**balls, the answer is easy as it is equivalent to the combinations of arranging N 0s and (M-1) 1s in a queue.$$\binom{M+N-1}{N}$$

However, if the boxes are themselves

**indistinguishable**(which I name them "piles" instead), how should I compute the number of combinations there? For example, I have 4 balls being distributed to 3 piles, the combinations should be (4,0,0), (3,1,0), (2,2,0), (2,1,1) only. So the number of ways = 4. I wonder if there is a simple way to compute the general case for M**indistinguishable**balls into N piles.