Factorial Calculator

Calculate factorials instantly for any non-negative integer. This factorial calculator provides both the exact result and scientific notation for large numbers, perfect for maths problems involving permutations, combinations, and probability calculations.

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Please enter a valid whole number between 0 and 170.

What is a Factorial?

A factorial is a mathematical function that multiplies a given number by every positive integer below it down to 1. The factorial of a number n is denoted by n! (pronounced “n factorial”). This operation is fundamental in combinatorics, probability theory, and various branches of mathematics.

Factorial Formula:

n! = n × (n - 1) × (n - 2) × (n - 3) × ... × 2 × 1

For example, 5! equals 5 × 4 × 3 × 2 × 1 = 120. By definition, 0! equals 1, which is crucial for many mathematical formulas and proofs.

How to Calculate Factorials

Calculating factorials involves multiplying a sequence of descending natural numbers. Here are the primary methods:

Manual Calculation

For small numbers, factorials can be calculated by hand. Simply multiply the number by each integer below it until you reach 1. For instance, to calculate 4!, multiply 4 × 3 × 2 × 1 = 24.

Calculator Method

For larger numbers, a scientific calculator or dedicated factorial calculator is essential. Most calculators have a factorial button (n!) or function that computes the result instantly, handling numbers up to several hundred.

Programming Approach

In programming, factorials are often calculated recursively or iteratively. The recursive method calls the function within itself, whilst the iterative method employs a loop to multiply consecutive integers.

Factorial Examples

Number (n)	Factorial Calculation	Result (n!)
0	By definition	1
1	1	1
3	3 × 2 × 1	6
5	5 × 4 × 3 × 2 × 1	120
7	7 × 6 × 5 × 4 × 3 × 2 × 1	5,040
10	10 × 9 × 8 × … × 1	3,628,800

Practical Applications of Factorials

Permutations and Arrangements

Factorials determine the number of ways to arrange objects in a specific order. For instance, there are 6! = 720 different ways to arrange six books on a shelf. This principle applies to scheduling, seating arrangements, and organisational planning.

Combinations and Selection

In combination problems, factorials help calculate how many ways you can select items from a larger set without regard to order. The formula C(n,r) = n! / (r!(n-r)!) is widely employed in lottery calculations, committee selections, and sampling methods.

Probability Calculations

Factorials are essential in probability theory, particularly when calculating the likelihood of specific arrangements or outcomes. They appear in distributions such as the Poisson distribution and in determining probabilities of random permutations.

Computer Science and Algorithms

In computer science, factorials are relevant for analysing algorithm complexity, particularly those involving sorting and searching operations. They also appear in recursive algorithms and data structure analysis.

Business and Logistics

Companies employ factorials to optimise logistics, such as determining delivery routes, allocating resources, and planning supply chains. These calculations help minimise costs and maximise efficiency in operations management.

Common Factorial Problems

Problem 1: Letter Arrangements

Question: How many different ways can the letters in the word “MATHS” be arranged?

Solution: Since there are 5 unique letters, calculate 5! = 5 × 4 × 3 × 2 × 1 = 120 different arrangements.

Problem 2: Seating Arrangements

Question: In how many ways can 8 people be seated in a row?

Solution: Calculate 8! = 40,320 different seating arrangements are possible.

Problem 3: Repeated Letters

Question: How many ways can you arrange the letters in “LETTER”?

Solution: There are 6 letters with two T’s and two E’s. Calculate 6! / (2! × 2!) = 720 / 4 = 180 arrangements.

Factorial Properties and Rules

Several key properties govern factorial operations:

Zero Factorial: 0! = 1 by mathematical definition, essential for formulas involving empty sets and base cases in recursion.
Recursive Property: n! = n × (n-1)!, which means each factorial builds upon the previous one.
Growth Rate: Factorials grow extremely rapidly. For instance, 20! exceeds 2 quintillion, whilst 100! contains 158 digits.
Non-negativity: Factorials are only defined for non-negative integers in standard mathematics.
Divisibility: n! is divisible by all integers from 1 to n.

Frequently Asked Questions

What is 0 factorial?

0! equals 1 by definition. This may seem counterintuitive, but it is mathematically necessary for consistency in formulas involving permutations and combinations. The reasoning is that there is exactly one way to arrange zero objects: do nothing.

Can you calculate factorials for negative numbers?

Standard factorials are not defined for negative integers. However, the gamma function extends the factorial concept to real and complex numbers, where Γ(n+1) = n! for non-negative integers.

What is the largest factorial a calculator can compute?

Most calculators can compute factorials up to approximately 170! before encountering overflow errors. The value 170! is roughly 7.26 × 10^306, which exceeds the limit of standard floating-point representation. Specialised software can handle larger factorials using arbitrary-precision arithmetic.

How are factorials written in mathematical notation?

Factorials are denoted by an exclamation mark (!) placed after the number. For example, 5 factorial is written as 5!. Double factorials (n!!) and multifactorials also exist for specialised applications.

What is the difference between permutations and combinations?

Permutations count arrangements where order matters, calculated as P(n,r) = n! / (n-r)!. Combinations count selections where order does not matter, calculated as C(n,r) = n! / (r!(n-r)!). Both formulae rely heavily on factorials.

Why do factorials grow so quickly?

Factorials exhibit exponential growth because each successive factorial multiplies by an increasingly larger number. This rapid expansion makes them invaluable for representing vast numbers of possibilities in combinatorial problems but also means they quickly exceed computational limits.

Can factorials be used for decimal numbers?

Whilst standard factorials are defined only for non-negative integers, the gamma function Γ(x) extends this concept to real and complex numbers. For a positive real number x, Γ(x+1) = x × Γ(x), and Γ(n) = (n-1)! for positive integers n.