Permutations & Combinations Calculator - nPr, nCr & Factorials

Calculate permutations (nPr) where order matters and combinations (nCr) where order does not matter, with or without repetition, plus factorials (n!).

⚙️ Configuration & Inputs

Total Number of Items (n) The total set size (e.g. 52 cards, 10 team members)

Number of Items to Choose (r) Sample subset size to arrange or select

📊 Results & Analysis

Combinations nCr (Order Ignored) 120 n! / (r! × (n - r)!)

Permutations nPr (Order Matters) 720 n! / (n - r)!

Factorial n! 3,628,800 Total ways to arrange all 10 items

Factorial r! 6 Arrangements of chosen subset

Permutations (With Repetition) 1,000 Formula: nʳ (e.g. lock pin codes)

Combinations (With Repetition) 220 Formula: (n + r - 1)! / (r!(n - 1)!)

Permutations vs. Combinations: What is the Difference?

The single most critical question in combinatorics is: Does the order of selection matter?

1. Permutations: Order Matters!

If you are choosing a 1st, 2nd, and 3rd place winner in a race, or picking a 4-digit security PIN, {1, 2, 3} is completely different from {3, 2, 1}.

nPr = n! / (n - r)!

2. Combinations: Order Does Not Matter!

If you are picking 3 people for a committee or dealt 5 cards in poker, {Alice, Bob, Charlie} is identical to {Charlie, Bob, Alice}.

nCr = C(n, r) = n! / [r! × (n - r)!] = nPr / r!

3. Combinatorics Quick Reference Summary

Type	Order Matters?	Repetition Allowed?	Formula
Permutation	YES	NO	`n! / (n - r)!`
Permutation w/ Repetition	YES	YES	`nʳ`
Combination	NO	NO	`n! / [r!(n - r)!]`
Combination w/ Repetition	NO	YES	`(n + r - 1)! / [r!(n - 1)!]`

Frequently Asked Questions

By definition, the factorial of n represents the number of ways to arrange n items. There is exactly one way to arrange zero items: the empty set { }. Mathematically, this also preserves the recursive formula: n! = n × (n - 1)!, which for n = 1 requires 1! = 1 × 0!, thus 0! must equal 1.

Because the order cards are dealt into your hand does not matter, this is a combination problem: 52 C 5 = 52! / (5! × 47!) = (52 × 51 × 50 × 49 × 48) / (5 × 4 × 3 × 2 × 1) = 2,598,960 possible hands.

A typical mechanical 'combination lock' is actually misnamed by everyday language: because the sequence of numbers matters (e.g. 24-10-38 is not the same as 38-10-24), it is technically a permutation lock!