Permutation & Combination Calculator

nPr and nCr, Solved Instantly and Side by Side

Permutations and combinations trip people up for the same reason every time: it's easy to forget which one cares about order. This calculator skips the mental gymnastics — enter n (your total pool of items) and r (how many you're choosing), and it shows you both P(n,r) and C(n,r) at once, along with the factorial formula behind each answer. It also handles arbitrarily large numbers without rounding errors, so n=50 or n=500 comes back exact, not an approximation.

How to use it

• Enter n, the total number of distinct items you're choosing from (a deck of cards, a list of runners, a set of passwords).
• Enter r, how many of those items you're actually selecting or arranging.
• Leave "allow repetition" unchecked for the classic case (each item can only be used once) — check it if items can repeat, like digits in a PIN code.
• Read both results at once: permutations (order matters, so ABC ≠ BCA) and combinations (order doesn't matter, so ABC = BCA), plus the exact formula worked out step by step.

Real-world use cases

Working out how many different 4-digit PIN codes exist (permutations with repetition), figuring out how many ways to award 1st/2nd/3rd place among 12 runners (permutations, no repetition), calculating how many possible 5-card poker hands exist from a 52-card deck (combinations), or checking a probability homework answer where you need to know if the problem wants an arrangement or just a selection.

Frequently asked questions

What's the actual difference between a permutation and a combination?

A permutation counts arrangements where order matters — picking a 1st, 2nd, and 3rd place winner from a group gives a different result depending on who comes first. A combination counts selections where order doesn't matter — picking a 3-person committee from a group gives the same result no matter what order you picked them in.

What does "with repetition" actually change?

Without repetition, once an item is chosen it's used up (like dealing cards from a deck). With repetition, an item can be picked more than once (like each digit in a phone number, which can repeat). Turning it on switches the formulas from n!/(n-r)! and n!/(r!(n-r)!) to n^r and C(n+r-1, r) respectively.

Why do combinations always come out smaller than permutations for the same n and r?

Every combination of r items can be arranged in r! different orders, and each of those orderings counts as a separate permutation. So permutations = combinations × r! — combinations is always the smaller number (or equal, when r is 0 or 1).

Can r be larger than n?

Only if repetition is allowed. Without repetition you can't select more distinct items than exist in the pool (you can't deal 10 unique cards from a 5-card hand), so the calculator flags that as invalid. With repetition, r can be any size since items can repeat.

How does this stay accurate for large numbers like 20! or 100!?

The calculator uses arbitrary-precision integer math instead of regular floating-point numbers, which lose precision past about 15-16 digits. That's why you'll see the exact digit count on very large results instead of a rounded scientific-notation approximation.