Factorial Formula

The factorial of a non-negative integer n, written n!, is the product of all positive integers from 1 to n: n!

The Formula

n!=n×(n−1)×(n−2)×⋯×2×1n! = n \times (n - 1) \times (n - 2) \times \cdots \times 2 \times 1

When to use: Factorial counts the number of ways to arrange nn distinct objects in a row — for 3 items, there are 3!=63! = 6 possible orderings.

Quick Example

5!=5×4×3×2×1=1205! = 5 \times 4 \times 3 \times 2 \times 1 = 120 0!=10! = 1 (by definition).

Notation

n!n! reads 'nn factorial'; by convention 0!=10! = 1

What This Formula Means

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

Factorial counts the number of ways to arrange nn distinct objects in a row — for 3 items, there are 3!=63! = 6 possible orderings.

Formal View

n!=∏k=1nkn! = \prod_{k=1}^{n} k for n≥1n \geq 1, with 0!=10! = 1 by convention; equivalently n!=n⋅(n−1)!n! = n \cdot (n-1)!

Worked Examples

Example 1

easy
Compute 7!7!.

Answer

7!=50407! = 5040

First step

1
Recall the factorial definition: n!=n×(n−1)×⋯×2×1n! = n \times (n-1) \times \cdots \times 2 \times 1. Write out 7!7!: 7!=7×6×5×4×3×2×17! = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1

Full solution

  1. 2
    Multiply step by step: 7×6=427 \times 6 = 42, then 42×5=21042 \times 5 = 210, then 210×4=840210 \times 4 = 840, then 840×3=2520840 \times 3 = 2520.
  2. 3
    Complete the product: 2520×2=50402520 \times 2 = 5040, so 7!=50407! = 5040.
The factorial n!n! is the product of all positive integers from 11 to nn. By convention, 0!=10! = 1. Factorials grow extremely fast.

Example 2

medium
Simplify 10!8!\frac{10!}{8!}.

Example 3

medium
Simplify (n+1)!(n−1)!\dfrac{(n+1)!}{(n-1)!}.

Common Mistakes

  • Setting 0!=00!=0 — by definition 0!=10!=1, which keeps the permutation/combination formulas consistent.
  • Multiplying only down to a wrong stopping point — go all the way to 11 (e.g. 4!=4⋅3⋅2⋅14!=4\cdot3\cdot2\cdot1, not 4⋅3⋅24\cdot3\cdot2).
  • Confusing n!n! with n2n^2 or 2n2n — 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!10! is over 3 million). Misremembering 0!=10!=1 quietly breaks those formulas. Recognizing it by "Am I counting the ways to arrange all nn distinct items, multiplying nn down to 11?" — 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 nn, written n!n!, is the product of all positive integers from 1 to nn: n!=n⋅(n−1)⋯2⋅1n! = n \cdot (n-1) \cdots 2 \cdot 1.

How do you use the Factorial formula?

Factorial counts the number of ways to arrange nn distinct objects in a row — for 3 items, there are 3!=63! = 6 possible orderings.

What do the symbols mean in the Factorial formula?

n!n! reads 'nn factorial'; by convention 0!=10! = 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!10! is over 3 million). Misremembering 0!=10!=1 quietly breaks those formulas. Recognizing it by "Am I counting the ways to arrange all nn distinct items, multiplying nn down to 11?" — 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!=00!=0. Asking "Am I counting the ways to arrange all nn distinct items, multiplying nn down to 11?" 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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