Factorial Calculator

Four calculations with different meanings

n! counts orderings of n distinct items. n!! multiplies every second positive integer. !n counts derangements—permutations where no item remains in its original position. Γ(z) extends the factorial relationship: Γ(n)=(n−1)! for positive integers.

Factorial and derangement modes use nonnegative whole numbers. The conventional values are 0!=1, 1!=1, !0=1 and !1=0. Double-factorial mode also permits −1, with (−1)!!=1. Gamma has poles at nonpositive integers and is evaluated by a numerical approximation elsewhere.

Reproduce the examples

For n=5, 5!=120, 5!!=5×3×1=15, and !5=44. For gamma, enter 5 to obtain Γ(5)=24, not 120. Entering 0.5 gives Γ(0.5)=√π≈1.772453851.

The integer calculations use BigInt. Large outputs are shown in shortened scientific notation with a digit count; the shortened mantissa does not display every exact digit. Factorial accepts up to 1000, while double factorial and derangements have smaller limits shown in validation messages.

Gamma uses finite floating-point arithmetic. Inputs with nonfinite output are rejected; this is not an arbitrary-precision gamma calculator. Its numerical approximation and the exact integer modes have different accuracy properties.

Interpretation and limits

A factorial is a count, not a probability. Rearranging repeated indistinguishable items needs a correction: arrangements of AAB total 3!/2!=3. More complex restrictions may require another counting argument.

Do not confuse Γ(n) with Γ(n+1)=n!, or double factorial with (n!)!. Mode changes clear the old result so its label cannot accidentally describe a different calculation.

Continue

Read counting principles and use the combinations/permutations tool for selections.