# Factorial Calculator

> Calculate the factorial of any number (n!). Solve permutations, combinations, and probability problems.

**Category:** Math
**Keywords:** factorial, n!, permutation, combination, probability, math
**URL:** https://complete.tools/factorial-calculator

## How it calculates

The factorial of a non-negative integer n is calculated using the formula: n! = n × (n - 1) × (n - 2) × ... × 3 × 2 × 1. The factorial of 0 is defined to be 1. In this formula, n represents the integer whose factorial is being calculated. For example, to compute 4!, substitute n with 4: 4! = 4 × 3 × 2 × 1 = 24. The relationship between factorials is essential in permutations and combinations. For permutations of r items from n items, the formula is P(n, r) = n! / (n - r)!. For combinations, the formula is C(n, r) = n! / [r! × (n - r)!]. Understanding these relationships is crucial in statistical analysis and combinatorial calculations.

## Who should use this

Mathematicians performing combinatorial analysis for research studies, computer scientists developing algorithms for sorting data, and statisticians conducting probability assessments in surveys are examples of professionals who can benefit from this tool. Additionally, educators designing curriculum materials for advanced mathematics courses may find it useful for demonstrating factorial concepts.

## Worked examples

Example 1: Calculating the factorial of 6. To find 6!, use the formula: 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. This value is applicable in various combinatorial problems, such as arranging 6 different books on a shelf. 

Example 2: Finding combinations. If a teacher wants to choose 3 students from a class of 10 for a project, the number of combinations can be calculated as C(10, 3) = 10! / [3! × (10 - 3)!] = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 120. This result indicates there are 120 different ways to select 3 students from 10.

## Limitations

This tool has several limitations. First, it can only handle non-negative integers, as factorials for negative integers are undefined. Second, calculating the factorial of large numbers (greater than 20) may lead to precision errors due to the rapid growth of factorial values beyond typical computational limits. Third, the tool assumes that inputs are integers; decimal or non-integer inputs will not yield valid results. Lastly, the calculator does not account for overflow errors in programming environments where integer limits are exceeded.

## FAQs

**Q:** How does the factorial function grow compared to polynomial functions?


**A:** Factorial functions grow significantly faster than polynomial functions. For example, while n^2 increases quadratically, n! grows exponentially, leading to much larger values for larger n.


**Q:** Can factorials be defined for non-integer values?


**A:** Yes, through the Gamma function, which extends the factorial to non-integer values, where n! = Γ(n + 1). This function is crucial in advanced mathematics, particularly in calculus.


**Q:** What practical applications do permutations and combinations have?


**A:** Permutations and combinations are widely used in fields such as cryptography for secure data transmission, operations research for optimizing resource allocation, and genetics for predicting trait outcomes in breeding experiments.


**Q:** How is the concept of factorial related to series expansions in calculus?


**A:** Factorials appear in the denominators of the terms in Taylor series expansions, such as e^x = Σ (x^n / n!) from n = 0 to infinity, illustrating their importance in mathematical analysis.

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