Free Permutation & Combination Calculator – nPr and nCr

Calculate Permutations (nPr) and Combinations (nCr)

The Helperzy Permutation and Combination Calculator computes both nPr and nCr from the two values n and r that you enter. A permutation counts how many ways you can arrange r items chosen from a set of n when the order matters — think of assigning first, second, and third place in a race. A combination counts how many ways you can choose r items when the order does not matter — think of selecting a committee where no one outranks another. Both results appear side by side so you can compare them directly and immediately see how removing the requirement of order affects the count — combinations are always smaller than or equal to permutations for the same n and r. The formulas rely on factorials. The permutation nPr equals n! divided by (n − r)!, which counts every ordered arrangement of r items from n. The combination nCr equals n! divided by [r! × (n − r)!], which takes the permutation count and divides out the r! duplicate orderings within each group. For example, choosing 2 items from 5: nPr = 5!/3! = 120/6 = 20 ordered arrangements, and nCr = 5!/(2! × 3!) = 120/(2 × 6) = 10 unordered selections. The ratio between a permutation count and a combination count is always exactly r!, because that is how many distinct orderings exist for any group of r items. Understanding this ratio helps you reason about whether a problem cares about order or not. A worked example with larger numbers: let n = 10 and r = 3. The permutation is 10 × 9 × 8 = 720, meaning there are 720 ways to assign gold, silver, and bronze medals among 10 athletes. The combination is 720 ÷ 6 = 120, meaning there are 120 ways to pick a 3-person committee from 10 people when rank does not matter. Enter these values and both answers appear instantly, confirming the 6-to-1 ratio that comes from 3! = 6. For a larger example, try n = 52 and r = 5 to find poker hand counts: 52C5 = 2,598,960 possible five-card hands from a standard deck, while 52P5 = 311,875,200 ordered sequences. These counts are the foundation of probability and statistics. They tell you how many possible outcomes exist, which forms the denominator in classical probability problems. They appear in lottery odds, card-hand calculations, password strength estimates, sampling designs in experiments, seating arrangements, tournament bracket counts, and any situation where you need to know how many distinct selections or orderings are possible from a given pool. Being able to compute them quickly makes the larger probability or statistics problem far more approachable, since the counting step involving large factorials is often exactly where students get stuck, lose confidence, or make arithmetic errors that cascade through the rest of the solution. The calculator guards against invalid input. The value r cannot exceed n, because you cannot choose or arrange more items than exist in the set, so the tool shows a clear message in that case. Both n and r must be non-negative whole numbers; negative values or decimals are flagged rather than producing a wrong answer. For very large n the exact factorial can exceed what a 64-bit number represents, and the calculator handles that gracefully with a clear overflow message. All computation happens in your browser with no server communication, so your numbers are never uploaded or stored. Note that these formulas assume selection without repetition — if items can be repeated, different counting rules apply, and this tool is specifically for the without-repetition case that dominates textbook and real-world problems.

Permutation & Combination Calculator Formula & Method

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