Compute the factorial of any non‑negative integer. Explore factorials in permutations, combinations, and series.
For a positive integer \( n \), the factorial \( n! \) (read "n factorial") is:
\[ n! = n \times (n-1) \times (n-2) \times \cdots \times 2 \times 1 \]
with the special case \( 0! = 1 \). Factorials grow very rapidly and appear in combinatorics, calculus, and many other areas.
Enter a non‑negative integer and click Calculate
For \( n \ge 22 \), results are shown in scientific notation (e.g., \( 22! = 1.1240007277776077 \times 10^{21} \)).
All examples below automatically update using your current value of \( n \).
The number of ways to arrange \( r \) items from \( n \) distinct items (order matters) is:
\[ P(n, r) = \frac{n!}{(n - r)!} \]
For \( r = 3 \), calculate \( P(n, 3) \) using your current \( n \):
| \( n \) | \( P(n, 3) = \dfrac{n!}{(n-3)!} \) | Numerical Value |
|---|---|---|
| 10 | \( \frac{10!}{7!} \) | — |
The number of ways to choose \( r \) items from \( n \) (order does not matter) is:
\[ C(n, r) = \frac{n!}{r! \, (n - r)!} \]
For \( r = 2 \), calculate \( C(n, 2) \) using your current \( n \):
| \( n \) | \( C(n, 2) = \dfrac{n!}{2! \, (n-2)!} \) | Numerical Value |
|---|---|---|
| 10 | \( \frac{10!}{2! \cdot 8!} \) | — |
The exponential function can be expressed as an infinite series using factorials:
\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \]
Using \( x = 1 \), the sum of the first \( n+1 \) terms approximates \( e \).
The partial sum approaches \( e \approx 2.718281828459045 \).