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Stirling's Approximation

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Tap a term to see what it means, or .

Calculate

Example: what is 10! by the approximation, and how close does it get to the exact value, 3,628,800? Change the value to work out your own case — try n = 1,000, a number with more than 2,500 digits.

Step by step

    What each term means

    Factorial (n!)
    The product 1 · 2 · 3 · … · n. It grows so fast that 171! is already beyond the largest number an ordinary calculator stores.
    Correction Factor (√(2πn))
    Adjusts the scale of the approximation. The π comes from the same integral that gives the √(2π) in the normal distribution.
    Main Term ((n/e)ⁿ)
    Carries almost all of the factorial's size. In logarithms it becomes n ln n − n, which is how statistical software handles large factorials.