Permutation and Combination Calculator
Calculate permutations (nPr) and combinations (nCr), with or without repetition, and list every arrangement. Free online calculator with formulas shown.
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Part of the toolkit
How to use Permutation and Combination Calculator
- 1Answer two questions: does the order matter, and can items repeat? This picks the right formula
- 2Enter n, the number of items to choose from, and r, how many you pick
- 3Read the exact count with the worked formula, compare all four cases, and list every arrangement for small sets
Frequently asked questions
What is the difference between a permutation and a combination?▾
In a permutation the order matters; in a combination it doesn't. Picking 2 of A, B, C, D, E as a president and vice-president gives 20 permutations (AB and BA differ), but picking 2 for a committee gives 10 combinations (AB and BA are the same pair).
What is the formula for combinations (nCr)?▾
C(n, r) = n! ÷ (r! × (n − r)!). For choosing 2 from 5: 5! ÷ (2! × 3!) = 120 ÷ (2 × 6) = 10. Many calculators and textbooks write it as “n choose r”.
What is the formula for permutations (nPr)?▾
P(n, r) = n! ÷ (n − r)!, which is n × (n − 1) × … for r factors. For the gold, silver and bronze medals among 10 runners: 10 × 9 × 8 = 720 possible podiums.
How many combinations does a 4-digit PIN have?▾
Digits can repeat and order matters, so it's 10⁴ = 10,000 possible PINs (0000 to 9999). With 6 digits it's 10⁶ = 1,000,000. That is a permutation with repetition, nʳ.
What are the odds of winning a 6/49 lottery?▾
Choosing 6 numbers from 49 where order doesn't matter gives C(49, 6) = 13,983,816 combinations, so one ticket has a 1 in 13,983,816 chance of matching all six.
How many 5-card poker hands are there?▾
C(52, 5) = 2,598,960 different hands, since the order you receive the cards doesn't matter. If order did matter it would be P(52, 5) = 311,875,200.
Embed Permutation and Combination Calculator on your website▾
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