A 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 definition, 0! = 1.

What is a double factorial?

The double factorial of n, written as n!!, is the product of all positive integers up to n that have the same parity as n. For odd n: n!! = n × (n-2) × (n-4) × ... × 3 × 1. For even n: n!! = n × (n-2) × (n-4) × ... × 4 × 2.

What is the difference between permutation and combination?

Permutation (nPr) counts arrangements where order matters: nPr = n!/(n-r)!. Combination (nCr) counts selections where order doesn't matter: nCr = n!/[r!(n-r)!]. For example, choosing 2 from 3 items: permutations = 6 (AB, BA, AC, CA, BC, CB), combinations = 3 (AB, AC, BC).

What is a subfactorial?

The subfactorial of n, written as !n, counts the number of derangements - permutations where no element appears in its original position. For example, !3 = 2 because for {1,2,3}, only {2,3,1} and {3,1,2} have no element in its original position.

Why is 0! equal to 1?

By convention, 0! = 1. This makes sense because there's exactly one way to arrange zero objects (do nothing). It also ensures the recursive formula n! = n × (n-1)! works correctly: 1! = 1 × 0! requires 0! = 1.

What is the largest factorial I can calculate?

This calculator supports factorials up to 170! because 171! exceeds the maximum value that can be stored in a standard floating-point number. 170! ≈ 7.26 × 10^306.

Factorial Calculator

Calculate factorial (n!), double factorial (n!!), permutations (nPr), and combinations (nCr). See step-by-step solutions with formulas and explanations for large numbers.

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exclamation Factorial Calculator

Calculation Type

n (0-170)

About This Mode

Factorial (n!): Product of all positive integers from 1 to n. Example: 5! = 5×4×3×2×1 = 120

calculate Result

Expression

Result

120

Step-by-Step

5! = 5 × 4 × 3 × 2 × 1

= 120

Formulas

n! = n × (n-1) × ... × 1

nPr = n! / (n-r)!

nCr = n! / [r!(n-r)!]

Input Values

Calculation Type

expand_more

Sequence End

Show Steps

Yes

lightbulb Tips

•0! = 1 by definition
•nPr = n!/(n-r)!
•nCr = n!/[r!(n-r)!]
•Max: 170! (float limit)

exclamation Reference

Common Factorials

0! = 16! = 720 1! = 17! = 5,040 2! = 28! = 40,320 3! = 69! = 362,880 4! = 2410! = 3,628,800 5! = 12020! = 2.4×10¹⁸

Key Formulas

n! = n × (n-1)!

nPr = n!/(n-r)!

nCr = n!/[r!(n-r)!]

How to Use This Calculator

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Select Calculation

Choose factorial, permutation, combination, or other types.

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Enter n Value

Input the number for factorial calculation (0-170).

input

Enter r (if needed)

For permutations/combinations, enter the r value.

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View Results

See the result with step-by-step calculation.

The Formula

Factorial multiplies all positive integers up to n. It's used in probability, combinatorics, and calculus.

n! = n × (n-1) × (n-2) × ... × 2 × 1

lightbulb Variables Explained

n! Factorial of n
nPr Permutations: n!/(n-r)!
nCr Combinations: n!/[r!(n-r)!]

tips_and_updates Pro Tips

0! = 1 by definition - this is essential for many mathematical formulas

Factorials grow extremely fast: 10! = 3,628,800 and 20! has 19 digits

For permutations (order matters): nPr = n!/(n-r)!

For combinations (order doesn't matter): nCr = n!/[r!(n-r)!]

Double factorial n!! multiplies every other number: 7!! = 7×5×3×1 = 105

Calculators typically support up to 170! due to floating-point limits

Factorial, Permutation & Combination Calculator

Calculate factorials (n!), double factorials (n!!), permutations (nPr), and combinations (nCr) with step-by-step solutions. Essential for probability and statistics.

What is Factorial?

Factorial of n (written as n!) is the product of all positive integers up to n. For example, 5! = 5×4×3×2×1 = 120. Factorials are fundamental in counting and probability.

Permutations vs Combinations

Permutations count arrangements where order matters (nPr). Combinations count selections where order doesn't matter (nCr). Use permutations for rankings, combinations for groups.