Stirling

The Stirling formula (18th century) is an approximation of the factorial values for large
numbers. It is built on the integral of the ln function, with bigO considerations
addedin. In effect, it becomes true with larger numbers of
instances. This is the nature of asymptotic relationships. Thus, it is used
in physics when large numbers of particles are at play.

The version of the formula typically used in applications is

${\displaystyle \ln n!=n\ln n-n+O(\ln n)}$

(in big O notation, as ${\displaystyle n\to \infty }$)

It can also be a quick approximation for the gamma function, but only in
cases where exactness is not the primary concern.  The more precise
formulation:

 

La formule de Stirling est utilisée partout où les valeurs exactes d'une faculté sont sans importance. En particulier, lors du calcul des informations d'un message et de l' entropie d' un ensemble statistique de sous-systèmes, la formule de Stirling apporte de grandes simplifications.

Wikipedia

https://www.freecodecamp.org/news/big-o-notation-simply-explained-with-illustrations-and-video-87d5a71c0174/

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The original formula by de Moivre: