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I wrote:

Simplify[Binomial[n, k] - Binomial[n - 1, k]]

I expected Mathematica to simplify this according to Pascal's identity to:

Binomial[n-1, k-1]

but it did not do this. Is there a way to tell it to use binomial coefficient identities in the simplification?

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This works

FunctionExpand[Binomial[n, k] - Binomial[n - 1, k]] // FullSimplify

$$ \frac{\Gamma (n)}{\Gamma (k) \Gamma (-k+n+1)} $$

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  • $\begingroup$ I suppose you could follow this with a replacement rule: /. Gamma[a_]/(Gamma[b_] Gamma[c_]) /; c == a - b + 1 :> Binomial[a - 1, b - 1] $\endgroup$ – Mr.Wizard Jul 24 '18 at 15:33
  • $\begingroup$ @Mr Wizard: Thank you for your valuable comment. $\endgroup$ – user64494 Jul 24 '18 at 15:45

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