# ${n\atopwithdelims( )r}$ 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 nchooseRIsAnInteger 2013-03-22 15:02:01 2013-03-22 15:02:01 matte (1858) matte (1858) 6 matte (1858) Theorem msc 11B65 msc 05A10