Permutations and Combinations Calculator
Written by Thierno Sadou Diallo, formula verified per our methodology • Last checked on 9/6/2026
The number of combinations is calculated with nCr = n! ÷ (r! × (n−r)!), and the number of permutations with nPr = n! ÷ (n−r)!. Choosing 3 elements out of 10 gives 120 possible combinations (order doesn't matter) and 720 permutations (order matters).
Explanation
The difference between a combination and a permutation comes down entirely to order: a combination counts the number of ways to choose r elements out of n without regard to their order (for example, drawing 3 cards from a deck, where the order of drawing doesn't matter for the resulting hand), while a permutation counts the number of ways to choose AND arrange them (for example, awarding a podium to 3 runners out of 10, where the finishing order changes the result). Since each combination of r elements can be reordered in r! different ways, the number of permutations is always equal to the number of combinations multiplied by r! — which is why nPr is always greater than or equal to nCr for the same n and r (equal only when r is 0 or 1). Both formulas rely on the factorial (n!, the product of all integers from 1 to n), which grows extremely fast: beyond n = 170, the factorial exceeds the largest value a computer can represent with standard precision.
Example: choosing 3 elements out of 10
Inputs
Total number of elements: 10. Number chosen: 3.
Calculation
Combinations = 10! ÷ (3! × 7!) = 3,628,800 ÷ (6 × 5,040) = 3,628,800 ÷ 30,240 = 120. Permutations = 10! ÷ 7! = 3,628,800 ÷ 5,040 = 720.
Result
There are 120 possible combinations, and 720 permutations if order matters.
Frequently asked questions
When should I use a combination instead of a permutation?
Use a combination when only the final result matters, with no notion of order or rank (forming a team of 5 players out of 20, drawing a hand of cards). Use a permutation when order or rank matters (assigning numbered seats, defining a ranking, creating a password where the order of characters matters).
What does a result of 0 mean?
It means it's impossible to choose more elements (r) than are available (n): for example, choosing 5 elements out of only 3 has no solution, which logically gives 0 combinations and 0 permutations.
Why is n limited to 170?
Because the factorial grows extremely fast: 170! is already a number with more than 300 digits, close to the limit of standard numeric precision (the IEEE 754 standard). Beyond 170, the calculation would exceed this limit and become uncomputable with a computer's usual precision.