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Simplify Binomial

This works:

FullSimplify[Binomial[m + 1, n + 1]/Binomial[m, n]]

(1 + m)/(1 + n)

but this doesn't:

FullSimplify[Binomial[m, n + 1]/Binomial[m, n]]

Binomial[m, 1 + n]/Binomial[m, n]

How do I get Mathematica to simplify the last result to (m-n)/n?


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