Factorial Calculator
A factorial calculator finds the factorial of a non-negative whole number. Factorials are used in mathematics, probability, statistics, permutations, combinations, and counting problems where the order or arrangement of objects matters.
What Is a Factorial?
The factorial of a positive whole number is the product of that number and every positive whole number below it. The factorial symbol is an exclamation mark !.
For example:
5! = 5 Γ 4 Γ 3 Γ 2 Γ 1 = 120
Therefore, the factorial of 5 is 120.
Factorial Formula
For a positive integer n, the factorial is defined as:
n! = n Γ (n β 1) Γ (n β 2) Γ ... Γ 2 Γ 1
There is one important special case: 0! = 1.
Factorial Examples
3! = 3 Γ 2 Γ 1 = 6
4! = 4 Γ 3 Γ 2 Γ 1 = 24
6! = 6 Γ 5 Γ 4 Γ 3 Γ 2 Γ 1 = 720
10! = 3,628,800
Common Factorial Values
| Number | Factorial |
|---|---|
| 0 | 1 |
| 1 | 1 |
| 2 | 2 |
| 3 | 6 |
| 4 | 24 |
| 5 | 120 |
| 6 | 720 |
| 7 | 5,040 |
| 8 | 40,320 |
| 9 | 362,880 |
| 10 | 3,628,800 |
How Factorials Are Used
Factorials are an important part of counting formulas. They are used in permutations to count arrangements and in combinations to count selections where order does not matter.
For example, the number of ways to arrange 4 different objects is:
4! = 24
Factorials are also used in probability calculations, statistical formulas, and mathematical series.
Factorials and Large Numbers
Factorials grow very quickly as the number increases. For example, 5! is only 120, but 10! is already 3,628,800. Larger factorials can contain many digits, making a calculator useful for obtaining an accurate result.
Important Rules
A standard factorial is normally defined for non-negative whole numbers. Negative integers do not have ordinary factorial values, and factorials of non-integer values require extensions such as the Gamma function rather than the basic factorial definition.
Common Mistakes to Avoid
Do not confuse n! with n Γ n. For example, 5! is 120, not 25. Remember that the factorial multiplies all positive whole numbers from the given number down to 1.
Also remember that 0! = 1, which is an important rule used in combinations and other mathematical formulas.
Quick Summary
A factorial is the product of a non-negative whole number and all the positive integers below it. The notation is n!, and the basic rule is n! = n Γ (n β 1) Γ ... Γ 2 Γ 1, with 0! = 1.