Factorial Calculator

10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3,628,800. This means there are 3,628,800 different ways to arrange 10 distinct objects in a line.

0! = 1 by mathematical convention based on the empty product principle. It also ensures that formulas work correctly: for example, the number of ways to choose 0 items from n is n!/(0! x n!) = 1, which requires 0! = 1. Without this convention, many combinatorics formulas would need special cases.

The combination formula is C(n, r) = n! / (r! x (n-r)!). To choose 3 people from a group of 10: C(10, 3) = 10! / (3! x 7!) = 3,628,800 / (6 x 5,040) = 120 different groups. Unlike permutations, combinations do not care about order.

Each new factor multiplies by an increasingly large number. 10! is 3.6 million, but 20! is 2.4 quintillion (2.4 x 10^18). By 100!, the result has 158 digits. This rapid growth is called "super-exponential" and is why factorial is used in computational complexity theory.

Permutations (order matters) use P(n,r) = n!/(n-r)!. Combinations (order does not matter) use C(n,r) = n!/(r!(n-r)!). Example: arranging 3 books from 5 has P(5,3) = 60 permutations, but choosing 3 books from 5 has C(5,3) = 10 combinations.

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