1.2 Binomial coefficients and factorials
The binomial coefficient \(\binom {n}{k}\), written ’C(n, k) in MathComp.
The factorial \(n!\), written n‘! in MathComp.
Let \(p\) be prime and \(n \in \mathbb {N}\). Then
\[ v_p(n!) = \sum _{k=1}^{n} \left\lfloor \frac{n}{p^k} \right\rfloor . \]
For \(k \leq n\),
\[ \binom {n}{k} \cdot k! \cdot (n-k)! = n!. \]
\((0 {\lt} \binom {n}{k}) \iff (k \leq n)\).
\(0 {\lt} n!\) for all \(n\).