Factorial Calculator
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Other tools you may find usefulFactorial in mathematics - definition, history and application
Factorial (symbol: exclamation mark n!) is a mathematical function defined for natural numbers that assigns a given numbernto the product of all positive natural numbers less than or equal ton. Although this concept had been known for millennia (particularly in Indian combinatorics for determining permutations), the modern exclamation mark notation was only introduced in 1808 by the French mathematician Christian Krampf. The factorial plays a key role in combinatorics (in calculating permutations, combinations and variations), probability theory, mathematical analysis (e.g. in Taylor's formula and power series), and in computer algorithms for assessing the computational complexity of programs.
Our free online factorial calculator allows you to instantly determine the factorial of any given natural number. The tool also handles very large values, showing the result in scientific notation.
Mathematical formula for factorial (n!)
For any positive natural numbern, the factorial is defined as the product:
n! = 1 × 2 × 3 × ... × (n - 1) × n
Recursively, the factorial can be defined using the relationship:
n! = n × (n - 1)!for n ≥ 1
In mathematics, there is also a very important definition for zero, which may seem illogical at first glance:
0! = 1
This assumption is necessary for many combinatorial formulas (e.g. Newton's symbol) to work correctly for extreme values.
Table of factorial values for the first natural numbers
Factorial values grow extremely quickly (this is the so-called factorial growth, faster even than exponential growth). The table below shows the values of n! for initial natural numbers:
| Natural number (n) | Factorial product expansion | Final result (n!) | Practical application in permutation |
|---|---|---|---|
| 0 | Conventional definition | 1 | Number of ways to order the empty set (exactly 1). |
| 1 | 1 | 1 | 1 item can only be arranged in 1 way. |
| 2 | 1 × 2 | 2 | Two elements (e.g. A, B) can be arranged in 2 ways: AB, BA. |
| 3 | 1 × 2 × 3 | 6 | The three elements can be arranged in 6 ways. |
| 4 | 1 × 2 × 3 × 4 | 24 | The four books on the shelf can be arranged in 24 ways. |
| 5 | 1 × 2 × 3 × 4 × 5 | 120 | Number of possible seats for 5 people at the table. |
| 6 | 1 × 2 × 3 × ... × 6 | 720 | 720 different permutations for a 6-element set. |
| 8 | 1 × 2 × 3 × ... × 8 | 40 320 | Number of permutations for 8 different characters. |
| 10 | 1 × 2 × 3 × ... × 10 | 3,628,800 | Number of combinations in some simple puzzle games. |
Why is the factorial growing so fast?
The phenomenon of rapid growth of the factorial value is best illustrated by a comparison. While 5! that's only 120, that's 15! is over 1.3 trillion (1,307,674,368,000). The factorial of 70 exceeds the computational capabilities of standard calculators and has over 100 digits. In computer science, algorithms whose execution time increases exponentially are considered extremely inefficient and impossible to apply to larger data sets (the so-called traveling salesman problem).
Frequently asked questions (FAQ)
What is a factorial?
The factorial of a given natural number n (denoted n!) is the product of all positive natural numbers from 1 to n.
Why is the factorial of zero (0!) 1?
This results from the combinatorial definition (there is exactly one way to order a set of size zero - that is, doing nothing) and the need to maintain mathematical consistency in formulas, e.g. for Newton's symbol.
How to calculate factorial of a negative number?
The classical factorial is not defined for negative numbers. However, there is a generalization of factorials to real and complex numbers in the form of the so-called the Gamma (Γ) function, which allows calculations beyond the set of natural numbers.
What is a double factorial?
The double factorial (denoted n!!) is the product of natural numbers with the same parity as n, not greater than n. For example: 5!! = 1 × 3 × 5 = 15, and 6!! = 2 × 4 × 6 = 48.
How to calculate factorial in Excel spreadsheet?
In Excel, there is a built-in FACT function for this purpose (e.g. entering=FACT(5)will return 120).