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Stirling Formula
Stirling formula or Stirling approximation is used to finding the approximate value of factorial of a given number ( n! or \(\Gamma \) (n) for n >> ). It was named after James Stirling. Stirling formula is a good approximation formula, it helps in finding the factorial of larger numbers easily and it leads to exacts results for small values of any number say 'n'. Stirling formula is also used for Gamma function and it is used in applied mathematics. Let us learn the Stirling formula along with a few solved examples. Let us learn the Stirling formula along with a few solved examples.
What Is Stirling Formula?
Stirling formula gives a value of factorial of a number very close to the real value of the factorial of the number, something close to less than 2% error. Stirling approximation is used to finding the approximate value of factorial of a given number 'n' and It was named after James Stirling. Stirling formula can be given as :
ln n! \(\approx\) n ln nn
or
n! = \(\sqrt ( 2 \times \pi \times n) (\dfrac{n}{e})^n\)
Let us see how to use the Stirling formula in the following solved examples section.
Solved Examples Using Stirling Formula

Example 1: Find the value of 5 factorial using Stirling formula.
Solution:
To Find: 5 factorial.
Using Stirling Formula
n! = \(\sqrt ( 2 \times \pi \times n) (\dfrac{n}{e})^n\)
5 fact = \(\sqrt ( 2 \times \pi \times 5) (\dfrac{5}{e})^5\)
= 118.019
Answer: The value of factorial 5 using the Stirling formula is 118.019. The Stirling formula gives a 1.66 percent error.

Example 2 : Find the value of 11 factorial using Stirling formula.
Solution:
To Find:11 factorial.
Using Stirling Formula
n! = \(\sqrt ( 2 \times \pi \times n) (\dfrac{n}{e})^n\)
11! = \(\sqrt ( 2 \times \pi \times 11) (\dfrac{11}{e})^11\)
= 39615625.05
Answer: The value of factorial 11 using the Stirling formula is 39615625.05. The Stirling formula gives a 1.66 percent error.
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