The permutation and combination calculator computes nPr (arrangements where order matters) and nCr (selections where order doesn't matter) with step-by-step factorial computation.
Permutation & Combination Calculator
How to Calculate Permutations and Combinations
Permutations and combinations count different ways to select items from a set. The key question: does order matter?
Permutations: Order Matters
nPr = n! / (n-r)!. For a 3-digit lock code from digits 1-9 (no repeats): 9P3 = 9×8×7 = 504 possible codes. For gold/silver/bronze from 8 runners: 8P3 = 8×7×6 = 336. Each different arrangement is a separate permutation.
Combinations: Order Doesn't Matter
nCr = n! / (r! × (n-r)!). Choosing 3 people from 10 for a committee: 10C3 = 120. The 5-card poker hand: 52C5 = 2,598,960 possible hands. The 6-ball lottery: 49C6 = 13,983,816 — your odds of winning are 1 in 13.98 million.
Relationship
nPr = nCr × r!. Because each combination of r items can be arranged in r! ways to give r! permutations. 10C3 × 3! = 120 × 6 = 720 = 10P3.
FAQ
What is the difference between permutations and combinations?
Permutations count arrangements where order matters. Combinations count selections where order doesn't. P(lock code 1-2-3 ≠ 3-2-1): use permutations. Choosing 3 toppings from 10 (order doesn't matter): use combinations. Rule: nPr = n! / (n-r)!, nCr = n! / (r! × (n-r)!).
How do you calculate 10C3 (10 choose 3)?
10C3 = 10! / (3! × 7!) = (10 × 9 × 8) / (3 × 2 × 1) = 720/6 = 120. There are 120 ways to choose 3 items from 10 when order doesn't matter. This is why lottery odds are so low: 49C6 = 13,983,816.
Is this calculator free?
Yes, completely free with no signup required. All calculations run in your browser.
Is my data private?
Yes. All calculations run locally. Nothing is transmitted.
What is a factorial?
n! = n × (n-1) × (n-2) × ... × 1. 5! = 5×4×3×2×1 = 120. 0! = 1 by definition. Factorials grow extremely fast: 20! ≈ 2.4 × 10¹⁸. Factorials appear in permutations, combinations, Taylor series, and probability distributions.