https://brilliant.org/wiki/integer-equations-star-and-bars/

https://brilliant.org/wiki/identical-objects-into-distinct-bins/

So, the total number of objects consists of the number of balls or start or else plus the number of bins -1 (which is the number of bars needed). Note, that it does not matter if we find all the ways we can choose the number of bars from the total number of objects or if we count all the ways we can choose the objects:

{n+k-1\choose{k-1}}={n+k-1\choose{n}}

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