sage: (n,k,j)=var('n,k,j')
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j),k,j+1,n)
0
sage: (n,j)=(5,3)
sage: sum(binomial(n,k)*binomial(k-1,j)*(-1)**(k-1-j) for k in range(j
+1,n+1))
1

The symbolic sum being 0 is only trivially valid for n<j+1.

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