Mathematics · Probability
Permutation & Combination Calculator (nPr, nCr)
Count the ways to arrange (permutations, nPr) or choose (combinations, nCr) r items from n — with or without repetition — using exact whole-number arithmetic for very large answers, the factorial formula worked through, and the odds of one particular outcome.
Your inputs
Quick start — try a typical case
Your result Live
Without Repetition
720 arrangements · 120 selections
Choosing 3 from 10
- Permutations (nPr) — order matters
- 720
- Combinations (nCr) — order doesn't matter
- 120
- Chance of One Particular Selection
- 1 in 120
Working
P: Step 1 · Arrangements
nPr = n! ÷ (n − r)! = 10! ÷ 7!
Expression becomes 720
C: Step 2 · Selections
nCr = n! ÷ (r! × (n − r)!) = 10! ÷ (3! × 7!)
Expression becomes 120
Permutations count orderings (ABC ≠ CBA); combinations count groups (ABC = CBA). nPr = nCr × r!.
How this is calculated
nPr = n! ÷ (n − r)! · nCr = n! ÷ (r!(n − r)!) · with repetition: nʳ and (n + r − 1)! ÷ (r!(n − 1)!)
- n!
- n × (n − 1) × … × 1, with 0! = 1
Use permutations when order matters — arranging books on a shelf, race finishing positions, PIN codes. Use combinations when it doesn't — picking a team or lottery numbers. With repetition, each pick can be any of the n items again. Results are exact even when they run to hundreds of digits.
Assumptions & sources
- n and r up to 5,000.
Calculator ID MATH-PERMCOMB-001 · version 1.0.0
Common questions
What's the difference between a permutation and a combination?
Order. Choosing 3 of 10 people for gold, silver and bronze is a permutation (720 ways); choosing 3 of 10 for a committee is a combination (120 ways).
How many 4-digit PINs are there?
Digits can repeat and order matters: 10⁴ = 10,000.
What are the odds of a 6-from-49 lottery?
49C6 = 13,983,816, so 1 in about 14 million.