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Factorial notation

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Factorial Notation

Factorial notation, written n!, means the product of a whole number n and every positive whole number below it, down to 1. Learn the definition with worked examples, why 0! equals 1 by convention, and how factorials are the building block for permutations and combinations.

What factorial notation means

Factorial notation uses an exclamation mark to represent the product of a whole number and every positive whole number below it. For a whole number n, n! (read "n factorial") means n × (n − 1) × (n − 2) × ... × 2 × 1.

Factorial notation: 5! 5! (5 factorial) equals 5 times 4 times 3 times 2 times 1, which equals 120. 5! = 5 × 4 × 3 × 2 × 1 5! = 120 multiply every whole number from 5 down to 1
5! = 5 × 4 × 3 × 2 × 1 = 120.

Worked examples

4! = 4 × 3 × 2 × 1 = 24. 3! = 3 × 2 × 1 = 6. Factorials grow very quickly — 10! is already 3,628,800 — because each new factor multiplies the whole product again.

A special case: 0!

By definition, 0! = 1, not 0. This might look strange, but it is the value that makes later formulas — especially permutations and combinations — work correctly in every case, including when there is nothing left to arrange.

Why factorial notation matters

Factorial notation is the building block for counting problems: how many ways can a set of items be arranged, or how many ways can a smaller group be chosen from a larger one. Both permutations and combinations are built directly from factorials, so understanding n! first makes those formulas much easier to follow.

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