Permutation & Combination
nPr and nCr — count ordered arrangements and unordered selections.
Combinations (nCr)
120
order doesn't matter
Permutations (nPr)
720
AI Breakdown & Smart Takeaway
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How the Permutation & Combination works
This Permutation & Combination calculator computes nPr (ordered arrangements) and nCr (unordered selections) instantly from any n and r values — ideal for students, teachers, statisticians, and anyone working through probability or combinatorics problems.
At its core, this tool solves two related but fundamentally different counting problems. A permutation (nPr) answers 'how many ways can I arrange r items from a set of n, where order matters?' A combination (nCr) answers 'how many ways can I choose r items from n, where order does not matter?' The distinction is critical: choosing the lottery numbers 3, 7, 14 is the same combination regardless of the order you pick them, but arranging gold, silver, and bronze medalists from a pool of 10 athletes produces a different result for each ordering.
The calculator uses factorial arithmetic under the hood. For permutations, it computes n! divided by (n−r)!, effectively counting every distinct sequence. For combinations, it divides that result further by r! to collapse all the orderings of the chosen group into a single count. Because factorials grow astronomically fast — 20! exceeds 2.4 quintillion — doing this by hand is error-prone for even moderate values of n, which is exactly where a calculator becomes indispensable.
Two key factors govern your result: the size of the total set (n) and the size of the selection (r). Increasing n always increases both nPr and nCr. Increasing r has a more nuanced effect on combinations: nCr is symmetric, meaning choosing 3 from 10 gives the same count as choosing 7 from 10, because every group of 3 you pick corresponds to a unique group of 7 you left behind. This symmetry property is a powerful sanity check — if your nCr answer for r items doesn't match the answer for (n−r) items, something has gone wrong.
A common mistake is applying the wrong formula to a problem. Ask yourself one question before calculating: does swapping the order of selected items produce a new, distinct outcome? If yes (passwords, race placements, seating arrangements), use nPr. If no (committee members, card hands, pizza toppings), use nCr. Another frequent error is forgetting that both formulas require r ≤ n — you cannot arrange more items than exist in the set. The calculator enforces this constraint automatically, flagging invalid inputs so you don't chase a nonsensical result.
Formula
nPr = n!/(n−r)! · nCr = n!/(r!(n−r)!)
Pro tips
- Before entering values, explicitly label whether your problem is order-sensitive (use nPr) or order-insensitive (use nCr) — this single decision determines which result is meaningful.
- Use the symmetry property of combinations as a built-in check: nCr(n, r) must always equal nCr(n, n−r). If you're choosing 2 items from 8, verify your answer matches choosing 6 from 8.
- For permutations of an entire set (arranging all n items), simply compute n! directly — nPr where r = n reduces to n!, and many calculators display this as a special case.
- When solving multi-stage problems (e.g., choose a president AND a separate committee from the same club), compute each stage separately using the appropriate nPr or nCr, then multiply the results together using the multiplication principle.
- Watch for 'with repetition' variants: the standard nPr and nCr assume no item is reused. If repetition is allowed (e.g., PIN codes with repeated digits), the formulas change to n^r for ordered arrangements and C(n+r−1, r) for unordered selections.
Key terms
- Permutation (nPr)
- — The number of distinct ordered arrangements of r items selected from a set of n items, where changing the order produces a different outcome.
- Combination (nCr)
- — The number of distinct unordered subsets of r items that can be chosen from a set of n items, where the order of selection is irrelevant.
- Factorial (n!)
- — The product of all positive integers from 1 up to n (e.g., 5! = 120), forming the mathematical foundation of both permutation and combination formulas.
- n (Population Size)
- — The total number of distinct items or elements available in the set from which selections or arrangements are made.
- r (Selection Size)
- — The number of items being chosen or arranged at one time, which must be less than or equal to n for the formulas to be valid.
- Combinatorics
- — The branch of mathematics concerned with counting, arranging, and selecting objects, of which permutations and combinations are the two most fundamental operations.