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The reason is that your expression is not equal to zero !!!

If you multiply your expression by $2^{(n+1)}$, and simplify the two bionmials in the middle, you get ${n+1\choose k} - {n\choose k} + {n \choose k-1}$

This can not be zero since ${n+1\choose k} + {~n \choose k-1}$ is usually bigger than ${n\choose k}$ (unless $n=0$ and $k>1$).

The reason is that your expression is not equal to zero !!!

If you multiply your expression by $2^{(n+1)}$, $2^{n+1}$, and simplify the two bionmials in the middle, you get ${n+1\choose k} - {n\choose k} + {n \choose k-1}$

This can not be zero since ${n+1\choose k} + {~n \choose k-1}$ is usually bigger than ${n\choose k}$ (unless $n=0$ and $k>1$).$k>n+1$).