Permutation and Combination Calculator
Combinations count selections where order doesn’t matter: choosing 3 from 10 gives 120 ways (720 if order matters).
120
- Permutations nPr (order matters)
- 720
- Permutations with repetition nʳ
- 1000
- Combinations with repetition (n + r − 1)Cr
- 220
Step-by-step solution
- nPr = n! ÷ (n − r)! = 720
- nCr = n! ÷ (r!(n − r)!) = nPr ÷ r! = 720 ÷ 6 = 120
Permutation and Combination Formula
nPr = n!/(n − r)!, nCr = n!/(r!(n − r)!)
nCr = nPr ÷ r!
Worked Example
Choosing 3 items from 10:
- nPr = n! ÷ (n − r)! = 720
- nCr = n! ÷ (r!(n − r)!) = nPr ÷ r! = 720 ÷ 6 = 120
Permutations with repetition nʳ: 1000 · Combinations with repetition (n + r − 1)Cr: 220
Test-day tip
ACT counting questions often combine steps; multiply the choices at each stage, and divide by r! only when the order inside a group doesn’t matter.
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Frequently asked questions
How do I know if order matters?
If swapping two chosen items gives a different outcome (president and vice-president, a PIN), use permutations; for teams or committees, use combinations.
What is the fundamental counting principle?
Multiply the number of choices at each step: 3 shirts and 4 pairs of trousers make 12 outfits.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Permutation and Combination Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/permutation-combination-calculator/
IEEE
Scientific Converters, "Permutation and Combination Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/permutation-combination-calculator/