Theorem 1   For $n\ge r \ge 0$ , the binomial coefficient $$ {n \choose r} $$ is an integer.

Proof. The proof is by induction on $n$ . For $n=0$ , the claim is clear. Thus, suppose the claim holds for $n\ge 1$ . For $r=1,\ldots, n$ , Pascal's rule gives $$ {n +1 \choose r} = {n\choose r} + {n\choose r-1}. $$ That is, ${n +1 \choose 1}, \ldots, {n+1 \choose n}$ are integers. Since $$ {n +1\choose 0} = 1, \quad {n +1\choose n+1} = 1 $$ the proof is complete.