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A bit different from the uestions already asked here. I have N balls and M distinct bins, I want to divide up to N of my balls to those bins (meaning either 0 balls, or only one ball ,...) How many ways are there?

My calculation so far:

if I want to divide exactly r balls to m bins than the formula is: M+r-1 choose M than I need to sum all of those calculations from 0 to N and that's where I get stuck, Can I get some help? I think the answer should be M+N choose M, so maybe someone can explain how he gets to this result in a different way.

BTW, the motivation for this question is calculating the dimension of RM[m,r] if it matters.

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    Try adding one more bin, the "didn't get put in any others" bin. Now you're distributing $N$ items to $M+1$ bins, so you get ${M+1} \choose N$. – 2017-01-28
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    right, that simple... tnx – 2017-01-28

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