$$n! \approx \sqrt{2\pi n} \left(\frac{n}{e}\right)^n$$
which gives a good estimate of the value of \(n!\) for large values of \(n\).
Stirling's Theorem:
$$ \lim\limits_{n \to \infty} \frac{n!}{n^n e^{-n}} = \sqrt{2\pi} $$
which is equivalent to the Stirling's approximation formula.
Stirling numbers are closely related to the Bell numbers, which count how many ways to partition a set.
Stirling numbers of the first kind count, while Stirling numbers of the second kind give signed counts of, the permutations in which exactly \(k\) elements of an \(n\)-permutation have smaller indices in the original ordering than they have in the permutation.
Stirling matrix:
The \(n\)-th Stirling matrix is a \(n \times n\) square matrix, denoted \(S_n\), whose elements \(s_{nk}\) are given by the Stirling numbers of the second kind.