Combination & Permutation Calculator

Combinatorics is the branch of mathematics concerned with counting, arranging, and selecting objects. Two of its most fundamental tools are combinations and permutations. A combination counts the number of ways to choose a group of items from a larger set when the order of selection does not matter. A permutation counts the number of ways to arrange items when order does matter.

Consider five coloured balls: red, blue, green, yellow, and orange. If you reach in and grab two balls from a bag, the pair {red, blue} is the same as {blue, red} — that is a combination. But if you are picking first place and second place in a race, red-then-blue is a different result from blue-then-red — that is a permutation. The only difference between the two scenarios is whether the sequence matters.

Both formulas are derived from the same building block: the factorial function. The factorial of a positive integer n, written n!, is the product of every whole number from 1 up to n. The definitions 0! = 1 and 1! = 1 are conventions that make the formulas work correctly at the boundary cases where r = 0 or r = n.

The formula for combinations is:

Where n is the total number of items in the pool and r is how many you are choosing. The denominator contains two factorials: r! accounts for all the orderings of the chosen group that should be treated as identical, and (n−r)! removes the unchosen items. Dividing by both collapses all duplicate orderings into a single outcome, which is exactly what a combination requires.

As an example: C(5,2) = 5! / (2! × 3!) = 120 / (2 × 6) = 120 / 12 = 10. There are 10 ways to pick 2 items from 5 when the order you pick them in does not change which group you end up with.

The formula for permutations is:

This is the same as the combination formula but without dividing by r!. That missing r! is the critical difference: permutations keep every distinct ordering as a separate outcome, while combinations collapse all orderings of the same group into one.

As an example: P(5,2) = 5! / (5−2)! = 120 / 6 = 20. There are 20 ordered pairs you can form by picking 2 items from 5. Notice that P(5,2) = 20 = 2 × C(5,2) = 2 × 10. The relationship P(n,r) = C(n,r) × r! always holds — permutations are combinations multiplied by the number of orderings of the chosen group.

The fastest way to decide which formula to use is to ask: does the order of selection change the outcome?

A popular memory trick: a "combination lock" is actually a permutation lock. The code 1-2-3 and the code 3-2-1 open different safes, so order is very much required. The name is a historical misnomer that has stuck for centuries.

Let us work through both for n = 10 and r = 3.

Combinations and permutations appear constantly in probability, statistics, game theory, and everyday life.

Mixing up nCr and nPr is the most frequent error. If you need ordered arrangements and accidentally use C(n,r), your answer will be too small by a factor of r!. Always ask "does the order of selection change the outcome?" before choosing a formula.

Forgetting that 0! = 1 leads to errors when r = 0 or r = n. When r = n, the denominator contains (n−r)! = 0! = 1, which is correct and finite — not zero. Plugging 0 into the denominator would cause a division-by-zero error.

Computing n! in full before dividing creates problems for large n. The factorial of 40 has 47 digits, making it impractical to compute by hand. The better approach is the multiplicative formula C(n,r) = (n × (n−1) × … × (n−r+1)) / r!, which works with much smaller intermediate values. This calculator uses exact big-integer arithmetic so results are always precise.