How to Use the Factorial Calculator
This calculator helps you quickly find the factorial of a non-negative whole number (n!). To use the calculator:
- Enter the value of n.
- Click Calculate.
The calculator instantly returns the exact value of n! and shows you how the value was calculated, step-by-step.
What Is a Factorial?
The factorial of a number is the product of that number and every positive whole number below it down to 1. It is represented by an exclamation mark (!) after the number.
For example, 8! means “8 factorial” and is calculated as follows:
8! = 8 × 7 × 6 ×5 × 4 × 3 × 2 × 1
= 40,320
Therefore, 8! = 40,320
Factorials are commonly used in probability, combinations, permutations, and other counting probability problems such as binomial probabilities.
Factorial Formula
For a positive whole number n, the factorial formula is:
n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1
Where:
- n is a non-negative whole number; and
- ! represents factorial.
The factorial can also be written recursively as: n! = n × (n − 1)!. For example, 6! = 6 × 5!
Since 5! = 120:
6! = 6 × 120
= 720
How to Calculate a Factorial
To calculate a factorial manually, multiply the chosen whole number by every positive whole number below it until you reach 1.
Example: Find 7!
By definition, the factorial formula is: n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1
Substituting the value into the formula, we get:
7! = 7 × 6 × 5 × 4 × 3 × 2 × 1
= 5,040
Therefore, 7! = 5,040
Common Factorial Values
The table below shows the factorial values from 0! to 10!, including how each factorial is calculated.
| Factorial | Calculation | Value |
|---|---|---|
| 0! | By definition | 1 |
| 1! | 1 | 1 |
| 2! | 2 × 1 | 2 |
| 3! | 3 × 2 × 1 | 6 |
| 4! | 4 × 3 × 2 × 1 | 24 |
| 5! | 5 × 4 × 3 × 2 × 1 | 120 |
| 6! | 6 × 5 × 4 × 3 × 2 × 1 | 720 |
| 7! | 7 × 6 × 5 × 4 × 3 × 2 × 1 | 5,040 |
| 8! | 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 | 40,320 |
| 9! | 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 | 362,880 |
| 10! | 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 | 3,628,800 |
Therefore:
- 3 factorial, 3! = 6
- 4 factorial, 4! = 24
- 5 factorial, 5! = 120
- 6 factorial, 6! = 720
- 7 factorial, 7! = 5,040
- 8 factorial, 8! = 40,320
- 9 factorial, 9! = 362,880
- 10 factorial, 10! = 3,628,800
Frequently Asked Questions
A factorial calculator finds n!, the product of a non-negative whole number and all positive whole numbers below it. Enter n to get the exact factorial value and a step-by-step calculation.
By definition, 0! = 1. Although there are no positive numbers to multiply when n = 0, defining 0! as 1 makes many counting and probability formulas work correctly.
52 factorial, written as 52!, is the product of all positive whole numbers from 52 down to 1. Thus, 52! = 52 × 51 × 50 × 49 × 48 × … × 5 × 4 × 3 × 2 × 1 = 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
In standard form, 52! ≈ 8.065817517 × 10⁶⁷
For a positive whole number n, the factorial formula is: n! = n × (n − 1) × (n − 2) × … × 3 × 2 × 1
The standard factorial n! is not defined for negative whole numbers. As such, this factorial calculator accepts only non-negative whole numbers:
