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What I'm tyring to show is the number of solutions to the equation of natural numbers;

$$ x_1 + x_2 + x_3 +\cdots + x_m = k $$ is equal to $$ \binom{m + k - 1} m $$

To be blunt, I have no idea where to start

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up vote 5 down vote accepted

Consider a string with $k$ symbols, which you need to partitition into $m$ regions, say by inserting $m-1$ separator characters "|". That gives your total string of length $k+m-1$, of which you need to pick where the $m-1$ separators go.

the total number of ways to do that is $k+m-1\choose{m-1}$

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thanks for the quick reply. it always amazes me how i can miss things like this – user45793 Oct 23 '12 at 21:53

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