Factorial Formula

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n! = n · (n-1) … 2 · 1.

The Formula

n!=n×(n−1)×(n−2)×⋯×2×1

When to use: Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3!=6 possible orderings.

Quick Example

5!=5×4×3×2×1=120 0!=1 (by definition).

Notation

n! reads 'n factorial'; by convention 0!=1

What This Formula Means

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n!=n⋅(n−1)⋯2⋅1.

Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3!=6 possible orderings.

Formal View

n!=∏k=1nk for n≥1, with 0!=1 by convention; equivalently n!=n⋅(n−1)!

Worked Examples

Example 1

easy
Compute 7!.

Answer

7!=5040

First step

1
Recall the factorial definition: n!=n×(n−1)×⋯×2×1. Write out 7!: 7!=7×6×5×4×3×2×1

Full solution

  1. 2
    Multiply step by step: 7×6=42, then 42×5=210, then 210×4=840, then 840×3=2520.
  2. 3
    Complete the product: 2520×2=5040, so 7!=5040.
The factorial n! is the product of all positive integers from 1 to n. By convention, 0!=1. Factorials grow extremely fast.

Example 2

medium
Simplify 10!8!.

Example 3

medium
Simplify (n+1)!(n−1)!.

Common Mistakes

  • Setting 0!=0 — by definition 0!=1, which keeps the permutation/combination formulas consistent.
  • Multiplying only down to a wrong stopping point — go all the way to 1 (e.g. 4!=4⋅3⋅2⋅1, not 4⋅3⋅2).
  • Confusing n! with n2 or 2n — factorial multiplies a descending run, not a square or a double.

Why This Formula Matters

Factorial is the atom of counting — every permutation and combination formula is built from factorials, and it explains the explosive growth of arrangements (10! is over 3 million). Misremembering 0!=1 quietly breaks those formulas. Recognizing it by "Am I counting the ways to arrange all n distinct items, multiplying n down to 1?" — rather than by familiar numbers — is what lets a student tell it apart from permutation and exponent and combination in a mixed problem set.

Frequently Asked Questions

What is the Factorial formula?

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n!=n⋅(n−1)⋯2⋅1.

How do you use the Factorial formula?

Factorial counts the number of ways to arrange n distinct objects in a row — for 3 items, there are 3!=6 possible orderings.

What do the symbols mean in the Factorial formula?

n! reads 'n factorial'; by convention 0!=1

Why is the Factorial formula important in Math?

Factorial is the atom of counting — every permutation and combination formula is built from factorials, and it explains the explosive growth of arrangements (10! is over 3 million). Misremembering 0!=1 quietly breaks those formulas. Recognizing it by "Am I counting the ways to arrange all n distinct items, multiplying n down to 1?" — rather than by familiar numbers — is what lets a student tell it apart from permutation and exponent and combination in a mixed problem set.

What do students get wrong about Factorial?

The procedure for factorial is the easy part; the trap is setting 0!=0. Asking "Am I counting the ways to arrange all n distinct items, multiplying n down to 1?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

What should I learn before the Factorial formula?

Before studying the Factorial formula, you should understand: multiplication.

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