Combinations, arrangements and permutations calculator
Calculate combinations C(n,r), arrangements P(n,r) and the factorial from n and r.
Use combination when order does not matter (e.g. lottery) and arrangement when it does (e.g. podium).
How the calculation works
The key difference is whether order matters.
Combination (order does NOT matter)
Choosing a group regardless of order — like lottery numbers.
C(n,r) = n! ÷ [r!·(n−r)!]
Arrangement (order matters)
Choosing and ordering — like a podium of 1st, 2nd and 3rd.
P(n,r) = n! ÷ (n−r)!
Step-by-step example
From 5 people, pick 2:
- Combination (a pair): C(5,2) = 10
- Arrangement (1st and 2nd): P(5,2) = 20
Examples
- C(60,6) = 50,063,860 (Mega-Sena combinations).
- Podium with 8 athletes: P(8,3) = 336 gold/silver/bronze arrangements.
Frequently asked questions
How many combinations of 6 from 60?
C(60,6) = 50,063,860 possible combinations.
Combination or arrangement?
Ask: does changing the order give a different result? If yes, arrangement; if not, combination.
What is a permutation?
It is the arrangement of ALL n elements: P(n) = n!. E.g. ordering 5 books = 120 ways.
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Updated on June 18, 2026 · by Rafael Rossi · Methodology & sources