Permutations Calculator

Why order changes the count

A permutation counts ordered arrangements, so picking the same r items in a different sequence counts as a separate outcome. The formula nPr = n! / (n-r)! works by taking all n! total orderings of the full set, then dividing out the (n-r)! orderings of the items you didn't select — because those unselected items' internal order doesn't matter to the arrangement you're counting.

Worked example: awarding three medals

Eight racers finish a race, and you want to know how many different ways gold, silver, and bronze can be awarded among them. That's 8P3, since which racer gets which specific medal matters — gold to Racer A and silver to Racer B is a different outcome than the reverse. Computing it: 8P3 = 8! / (8-3)! = 8! / 5! = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (5 × 4 × 3 × 2 × 1). The 5! term cancels out everything below 6, leaving 8 × 7 × 6 = 336. There are 336 distinct ways to hand out the three medals.

How the calculator avoids overflow

Rather than computing 8! and 5! as separate huge numbers and dividing (which is what the calculation steps display for clarity), the underlying result is mathematically just the product of n counting down r terms: n × (n-1) × ... × (n-r+1). For large n this matters — the calculator computes full factorials for the step-by-step display, so very large values of n can produce enormous intermediate numbers even though the final ratio is comparatively small.

When to reach for permutations instead of combinations

Use this calculator whenever a role, rank, or position is being assigned — race placements, ordered passwords from a fixed character set, or assigning distinct job titles to a shortlist of candidates. If the items being chosen are functionally interchangeable (a committee with no distinct roles, for example), that's a combination question instead, which counts fewer outcomes because it ignores order.