Factorial Calculator


An online calculator to calculate the factorial of a positive integer.

Definition of the Factorial of a Positive Integer

If n is a positive integer, then the factorial of n written as \( n! \) (read as "n factorial") is given by
\( n ! = n \times (n-1) \times (n-2)....2 \times 1 \)
with \( 0! = 1\).
Example 1
\( 2! = 2 \times (2 - 1) = 2 \times 1 = 2 \)
\( 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3628800 \)

The factorial of positive integers increases very quickly.
Example 2
\( 40! = 8.159152832478977\times 10^{47} \)
\( 100! = 9.33262154439441 \times 10^{157} \)
Note: the powers of 10 are written with an "E", see example below.

Factorial Calculator of a Positive Integer

1 - Enter as a positive integer n or 0 and press "Calculate n!".
Note for n = 22 or larger, the results is written in scientific notation using "E" instead of 10.
Example: \( 22! = 1.1240007277776077e+21\)     is     \( 1.1240007277776077 10^{21} \)

n =            


Applications of Factorial in Mathematics

Factorials are used in the formulas of permutations and combinations.
Example 3

Factorials are used in the calculation of combinations. The combination of n objects taken r objects at a time is written as \( C(n,r) \) and is given in terms of factorials by the formula
\( C(n,r) = \dfrac{n!}{(n - r)! r!}\)

Example 4

Factorials are used in the calculation of permutations. The permutation of n objects taken r objects at a time, where order is important, is written as \( P(n,r) \) and is given in terms of factorials by the formula
\( P(n,r) = \dfrac{n!}{(n - r)}\)

Example 5

Factorials are used in series of functions in calculus and in turn these series are used in electronic calculators to compute functions such sin(x), cos(x), ln(x), ex. We list here some examples of functions given by series.
a) \( e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + ... \)
b) \( sin(x) = x - x^3/3! + x^5/5! + ... \)
c) \( cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ... \)

More References and Links

Factorial Questions with Solutions. Tutorials on evaluating and simplifying expressions with factorial notation.
Permutations and Combinations Problems.
Math Calculators and Solvers.

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