${n\\atopwithdelims( )r}$ is an integer

Theorem 1.

For $n\\geq r\\geq 0$ , the binomial coefficient

{n\\choose r}

is an integer.

Proof.

The proof is by induction on $n$ . For $n=0$ , theclaim is clear. Thus, suppose the claim holds for $n\\geq 1$ .For $r=1,\\ldots,n$ , Pascal’s rule gives

{n+1\\choose r}={n\\choose r}+{n\\choose r-1}.

That is, ${n+1\\choose 1},\\ldots,{n+1\\choose n}$ areintegers.Since

{n+1\\choose 0}=1,\\quad{n+1\\choose n+1}=1

the proof is complete.∎

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(nr) is an integer

Theorem 1.

Proof.

${n\\atopwithdelims( )r}$ is an integer