Stirling's Approximation in Puzzles

Computing something like "how many ways can you shuffle 52 cards" or estimating a binomial coefficient with large numbers means dealing with factorials, which grow so fast that exact computation by hand is impractical and even a calculator's output is hard to interpret intuitively. Stirling's approximation gives a simple estimate: $n! \approx \sqrt{2\pi n}\,(n/e)^n$, which becomes very accurate as $n$ gets large. In plain terms, a factorial's size is dominated by $n$ raised to roughly its own power divided by $e$, with a modest correction factor out front.

In an interview setting, candidates rarely need the full formula — the key trick is the simplified log version: $\ln(n!) \approx n\ln n - n$. This lets you estimate things like the order of magnitude of $100!$ mentally: $\ln(100!) \approx 100\ln100 - 100 \approx 100(4.6) - 100 = 360$, so $100! \approx e^{360}$, an astronomically large number confirmed to be roughly correct order-of-magnitude without ever multiplying 100 numbers together. This kind of quick estimation is exactly what interviewers probing for numerical intuition are testing.