Permutations & Combinations Calculator
Free nPr and nCr calculator — exact permutation and combination counts with formulas, for probability and counting problems.
Everyday Uses
Lottery odds
Compute 49C6 yourself and see exactly why the jackpot is so hard to hit.
Password and PIN counting
How many 4-digit PINs without repeats? Arrangement counting made exact.
Probability homework
Exact nPr and nCr values with big-integer precision — no scientific-notation rounding errors.
Team and committee counts
How many distinct groups of 4 from a class of 20? One input away.
Frequently Asked Questions
What is the difference between a permutation and a combination?
Permutations count arrangements where order matters; combinations count selections where it doesn't. Picking a president and vice-president from 10 people: 10P2 = 90 (order matters). Picking 2 committee members: 10C2 = 45 (order doesn't). Every combination corresponds to r! permutations, which is why nPr = nCr × r!.
How are nPr and nCr calculated?
nPr = n!/(n−r)! — multiply n × (n−1) × … for r factors. nCr = n!/(r!(n−r)!) — the same, divided by r! to cancel the orderings. Example: 10P3 = 10×9×8 = 720; 10C3 = 720/6 = 120. This calculator computes both exactly using big-integer arithmetic, so even large results are precise.
When do I use each in probability problems?
Use combinations for lottery-style problems ("chance of matching 6 of 49 numbers" → 49C6), card hands, and group selection. Use permutations when positions are distinct — race finishing orders, PINs without repeated digits, seating arrangements. The classic test: if swapping two chosen items creates a "different" outcome, order matters — use permutations.
What are 0! and nC0?
By definition 0! = 1, which makes the formulas consistent: nC0 = 1 (one way to choose nothing) and nCn = 1 (one way to choose everything). Similarly nP0 = 1. These edge cases aren't arbitrary — they're required for the binomial theorem and recursive identities to hold.