TITLE: $\binom{n}{r} = 8$ Is there any way to find such $n$ and $r$?
QUESTION [5 upvotes]: Let ${{n} \choose {r}} = 8$.
Is there any other choice of $n$ and $r$ except $8$ and $1$, $8$ and $7$
?
In general how to check that existence is guaranteed or not?

REPLY [2 votes]: $\binom{n}{r}\geq n$ as soon as $r\neq 0,n$. So the only way to have $\binom{n}{r}=8$ is for $n\leq 8$. A quick check (on a Pascal triangle) shows that the only solution is $\binom{8}{1}=\binom{8}{7}$