Permutation & Combination Calculator — nPr & nCr
Enter n (total items) and r (items chosen) to instantly calculate both nPr (ordered arrangements) and nCr (unordered selections).
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🔢 Result
Disclaimer: This tool performs standard mathematical calculations for educational and reference purposes. Always double-check results independently for graded coursework, exams, or safety-critical applications.
Permutations vs. Combinations
Both permutations and combinations count the number of ways to choose r items out of a total of n, but they answer slightly different questions. A permutation (written nPr) counts ordered arrangements — it matters which item comes first, second, third, and so on. A combination (written nCr) counts unordered selections — only which items are chosen matters, not the order they're picked in.
The Formulas
nPr = n! / (n − r)! — the number of ways to arrange r items out of n, where order counts.
nCr = n! / (r! × (n − r)!) — the number of ways to choose r items out of n, where order doesn't count. Equivalently, nCr = nPr ÷ r!, since dividing out the r! ways to reorder any chosen group collapses permutations down to combinations.
Worked Example
Suppose n = 5 and r = 3 — picking 3 items from a group of 5.
- nPr: 5 × 4 × 3 = 60 ordered arrangements (multiplying the 3 numbers counting down from 5)
- r! (3!): 3 × 2 × 1 = 6
- nCr: 60 ÷ 6 = 10 unordered selections
So there are 60 ways to arrange 3 items from a group of 5 in order, but only 10 distinct groups of 3 if order doesn't matter.