# 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 \( 1.1240007277776077× 10^{21} \)

## 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

\( C(n,r) = \dfrac{n!}{(n - r)! r!}\)

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

\( P(n,r) = \dfrac{n!}{(n - r)}\)

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), e^{x}. 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 .