${n\atopwithdelims( )r}$ 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 $n=0$ , the claim 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, ${n+1\choose 1},\ldots,{n+1\choose n}$ are integers. Since

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

the proof is complete. ∎

Title	${n\atopwithdelims( )r}$ is an integer
Canonical name	nchooseRIsAnInteger
Date of creation	2013-03-22 15:02:01
Last modified on	2013-03-22 15:02:01
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	6
Author	matte (1858)
Entry type	Theorem
Classification	msc 11B65
Classification	msc 05A10

(nr) is an integer

Theorem 1.

Proof.

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