How Big Are Factorials? A Simple Estimate for 52!
Eli Bendersky explores how to estimate the number of digits in a factorial without a calculator. Using Stirling's approximation, he shows that 52! has about 68 digits, and derives the formula from the Gamma function and Laplace's method. The post includes a practical approximation and the mathematical background, making it accessible for curious readers.
The real answer is 68, so this is very close! In estimates like this - when you're dealing with enormous numbers - being off by a couple of digits usually isn't a big deal.
- Sharlin
A quick and dirty approximation of the number of digits in n! is n lg n, which approximates n! from above, via the inequality
1 * 2 * … * n ≤ n * … * n.
(This approximation should be familiar to many from their algorithmics class.)
For a tighter bound, use n lg n - n/2, or a better approximation of ln 10 in place of 1/2 if you wish. This comes from Stirling's approximation which notes that
ln n! = n ln n - n + O(ln n).
- ninju
The author's casual mention of 52! at the opening of the article triggered an OLD webpage that I saw many years ago
https://czep.net/weblog/52cards.html
Anyone know how to determine the age of this page (it's got be at least 20yrs old)
- movpasd
Stirling's approximation is also used a lot in statistical mechanics, because you often have to calculate logs of state space sizes, which means lots of combinatorics and thus lots of factorials. Plus it's continuous so you can do calculus.