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I am trying to prove algebraically and using combinatorics that

$$\sum_{i=0}^r \binom{i + m -1}{m - 1} = \binom{m + r}{m}$$ where $m\geq1$ and $r\geq0$

The algebraic proof went smoothly but I am confused as to what the left hand side means. I understand the right hand side is showing how to choose $m$ elements out of $m+r$ elements, but how does that relate to the left hand side?

lulu
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Andrew Dews
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    Formatting note: use \sum instead of \sigma for a better display. – lulu Sep 26 '22 at 17:31
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    To your question, Hint: to specify $m$ choices out of $r+m$ options, you just need to choose $m-1$ out of the options that precede the last one – lulu Sep 26 '22 at 17:32
  • Do you know that the number of ways to solve $n=x_1+\cdots+x_m$ where the $x_i$ are positive integers is $\binom{n+m-1}{m-1}?$ This is usually proved using the "stars and bars" technique. https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)?wprov=sfti1 – Thomas Andrews Sep 26 '22 at 17:32
  • So the right side is the number of ways to write $r$ and the sum of $m+1$ numbers, and the left side is the number of ways to write all numbers $\leq r$ as the sum of $m$ numbers. – Thomas Andrews Sep 26 '22 at 17:35

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