Fixed-cardinality binomial coefficient asymptotic ($\left(\genfrac{}{}{0ex}{}{n}{q}\right)\sim n^q/q!$)

For a fixed nonnegative integer $q$, the number of $q$-element subsets of an $n$-element set satisfiesFor $q\ge 1$, divide the product formula for the binomial coefficient by $n^q$:The empty product gives the same result for $q=0$. For an explicit big theta notation bound, $n\ge 2q$ gives $n^q/(2^{q}q!)\le (\genfrac{}{}{0ex}{}{n}{q})\le n^q/q!$. Fixing $q$ is essential: these constants are not uniform when the subset size itself increases with $n$.