Simple one (Lucas theorem)

Consider the integers $${n \choose k}$$
for $k=0,1,\ldots,n$. It turns out that the number of those which are odd is always a power of $2$.

Proof. Direct application of Lucas theorem. $ \displaystyle {n \choose k}$ is odd exactly when the $1's$ in the binary represenation of $k$ correspond only to $1's$ in the binary representation of $n$. If $n$ has exactly $d$ $1's$ then the number of such $k$ is $2^d$.

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