Factorial Calculator

Calculate factorials, permutations, and combinations. Essential for probability and combinatorics problems.

Calculation Mode

Factorial (n!)

Permutation (nPr)

Combination (nCr)

Value of n

About Factorial Calculator

Understanding Factorials

The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! is defined as 1. Factorials grow extremely quickly — 20! already exceeds 2.4 quintillion — and are foundational to counting, probability, and combinatorics.

Permutations and Combinations

Permutations P(n, r)

A permutation counts the number of ways to arrange r items chosen from n distinct items where order matters.

P(n, r) = n! / (n − r)!

Combinations C(n, r)

A combination counts the number of ways to choose r items from n distinct items where order does not matter.

C(n, r) = n! / (r! × (n − r)!)

Key Formulas and Properties

0! = 1 (by definition)
n! = n × (n − 1)!
P(n, n) = n! (arranging all items)
P(n, 1) = n
C(n, 0) = C(n, n) = 1
C(n, r) = C(n, n − r) (symmetry property)
The number of permutations is always greater than or equal to the number of combinations for the same n and r

Real-World Applications

Probability

Calculating lottery odds and card game probabilities
Determining possible outcomes in experiments
Risk assessment and statistical analysis

Computer Science

Algorithm complexity analysis (e.g., O(n!) for brute-force)
Hashing and cryptographic functions
Generating permutations and combinations programmatically

Everyday Decisions

Seating arrangements and scheduling
Choosing teams or committees from a group
Password and PIN combination possibilities

Frequently Asked Questions

What is a factorial?

A factorial, written as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! equals 1.

What is the difference between a permutation and a combination?

A permutation counts arrangements where order matters (e.g., ranking winners 1st, 2nd, 3rd), while a combination counts selections where order does not matter (e.g., choosing 3 people for a committee). Permutations always produce a number greater than or equal to combinations for the same n and r.

Why is 0! equal to 1?

0! = 1 by definition because there is exactly one way to arrange zero items — doing nothing. This convention also ensures that formulas like C(n, 0) = n! / (0! × n!) = 1 work correctly.

How large can the input be?

This calculator supports values of n up to 170. Beyond that, the numbers become astronomically large. Internally the calculator uses BigInt arithmetic to handle values of n greater than 20, which exceed the safe integer limit for standard JavaScript numbers.

When would I use permutations vs. combinations?

Use permutations when the order of selection matters, such as assigning ranked positions, creating passwords, or arranging items in a sequence. Use combinations when order does not matter, such as choosing lottery numbers, forming committees, or selecting items from a menu.

What is the formula for combinations?

The combination formula is C(n, r) = n! / (r! × (n − r)!). It calculates how many ways you can choose r items from n items without regard to order. A useful property is that C(n, r) = C(n, n − r).

Additional Resources

Factorial - Wikipedia Combinations and Permutations - Khan Academy Permutation - Wikipedia

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