# Binomial sum identities

How do you persuade Mathematica to give simple identities involving binomial sums. For example

Sum[Binomial[a,i]*Binomial[b,i],{i,0,n}]

where n is bigger than both a and b.

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In[1]:= $Assumptions = n > a > 0 && n > b > 0 && a \[Element] Integers && b \[Element] Integers; In[2]:= Sum[Binomial[a, i]*Binomial[b, i], {i, 0, n}] Out[2]= Gamma[1 + a + b]/(Gamma[1 + a] Gamma[1 + b]) In[3]:= FullSimplify[%] Out[3]= (a + b)!/(a! b!) Note that it does give a generally valid result without these assumptions too. Here the assumptions helped us obtain a simpler result. I used the$Assumptions global variable here to avoid typing the assumptions twice, for Sum and FullSimplify.