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How do I algebraically $n{m+n \choose m}=(m+1){m+1 \choose n+1}$? I got the combinatorial proof on how to prove this but not sure where to start with this algebraically.

k.stm
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Dan986
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    Are these binomial coefficients? If so, the Latex code is n {m + n \choose m} = (m + 1) {m + 1 \choose n + 1}; put this between dollar signs $. –  Mar 13 '14 at 22:14
  • Duplicate: http://math.stackexchange.com/questions/710338/solve-algebraically-n-binommnm-m1-binom-mnm1 – anon Mar 13 '14 at 22:17

1 Answers1

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You might be having trouble because it's false (for example: if $n>m$, the right-hand side of your equation is zero while the left-hand side isn't). Perhaps you meant the following?:

$$(m+1){m\choose n} = (n+1) {{m+1}\choose{n+1}}$$

You can quickly show this algebraically by writing the binomial coefficients in terms of factorials.

EDIT: Aha, from the link sea turtles gave, I see that you might also mean this one:

$$n \binom{m+n}{m} = (m+1)\binom {m+n}{m+1}$$

This also follows as soon as you know how to write binomial coefficients in terms of factorials.

Andrew Dudzik
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