Stirling's Approximation

Factorials grow so fast that computing $n!$ directly for anything beyond small $n$ is impractical by hand, yet factorials show up constantly in combinatorics and probability — counting arrangements, computing binomial coefficients, evaluating tail probabilities. Stirling's approximation gives a closed-form estimate of $n!$ that's accurate enough for essentially all practical purposes, without multiplying $n$ terms together.

The formula states $n! \approx \sqrt{2\pi n} \left(\dfrac{n}{e}\right)^n$, where $e$ is Euler's number. In words: $n!$ behaves, for large $n$, like a large exponential term $(n/e)^n$ scaled by a comparatively gentle square-root correction factor $\sqrt{2\pi n}$. The exponential term dominates the size of the answer; the square-root term refines it.

For $n = 10$: the true value is $10! = 3{,}628{,}800$. Stirling's estimate gives $\sqrt{2\pi \times 10} \times (10/e)^{10} \approx 7.927 \times (3.6788)^{10} \approx 3{,}598{,}696$ — within about 0.8% of the exact answer, and the relative error shrinks further as $n$ grows.

A common working shortcut in mental-math or interview settings is the simpler bound $\ln(n!) \approx n\ln n - n$, dropping the square-root correction entirely — useful for quickly estimating the order of magnitude or the log of a factorial (as needed, for example, in Stirling-based approximations to binomial coefficients) when a rough answer is all that's required.