= Fixed-cardinality binomial coefficient asymptotic
{title2=$\binom nq\sim n^q/q!$}
For a fixed nonnegative <integer> $q$, the number of $q$-element <subsets> of an $n$-element <set> satisfies
$$
\binom nq\sim\frac{n^q}{q!}\quad(n\to\infty),
\qquad\binom nq=\Theta(n^q).
$$
For $q\geq1$, divide the product formula for the <binomial coefficient> by $n^q$:
$$
\frac{\binom nq}{n^q}=\frac1{q!}\prod_{j=0}^{q-1}\left(1-\frac jn\right)\longrightarrow\frac1{q!}.
$$
The empty product gives the same result for $q=0$. For an explicit <big theta notation> bound, $n\geq2q$ gives $n^q/(2^q q!)\leq\binom nq\leq n^q/q!$. Fixing $q$ is essential: these constants are not uniform when the subset size itself increases with $n$.
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