Permutation and Combination Calculator

This permutation and combination calculator computes both nPr (permutations) and nCr (combinations) in a dual-result dashboard. Enter total items (n) and items to choose (r); the tool validates r ≤ n when repetition is off and supports repetition allowed for advanced combinatorics. The Logic Visualizer shows how choices multiply (e.g., 10×9×8 for P(10,3)), and the Logic Trace breaks down factorial steps with comma-separated integers for values ≤ 1,000,000. Formatting options (scientific notation, significant figures) handle large factorials. All calculations use BigInt for exact arithmetic, no floating-point roundoff.

Permutations count arrangements where order matters. Picking a President and VP from 5 candidates gives P(5,2) = 5×4 = 20, AB and BA are different outcomes. Combinations count selections where order does not matter. Picking 2 committee members from 5 gives C(5,2) = 10, AB and BA are the same committee. The formulas reflect this: P(n,r) = n!/(n−r)! and C(n,r) = n!/(r!(n−r)!). Since C divides by r!, combinations are always ≤ permutations for the same n and r. Use this calculator to compare both and choose the correct formula for your problem.

Without repetition: P(n,r) = n!/(n−r)!; C(n,r) = n!/(r!(n−r)!). Requires r ≤ n. With repetition allowed: Permutations = n^r (each of r slots can be any of n items). Combinations with repetition use the stars-and-bars formula: C(n+r−1, r). The calculator switches formulas automatically based on the Repetition toggle and shows the appropriate Logic Trace. This covers the full range of combinatorics problems, from basic "how many ways to arrange" to advanced "combinations with replacement."

Enter the total number of items in your set as n (for example, 10 candidates) and the number you want to choose or arrange as r (for example, 3 positions). The calculator simultaneously displays both P(n,r) (permutations, order matters) and C(n,r) (combinations, order does not matter), each tagged so you can confirm which formula applies to your problem. If items can repeat, such as digits in a PIN code or letters in a password, enable the Repetition Allowed toggle; formulas switch to n^r for permutations and the stars-and-bars formula C(n+r−1, r) for combinations. Without repetition, r must be less than or equal to n. The Logic Visualizer shows r slots with the number of choices at each step, and the Logic Trace breaks down every factorial operation. Use Copy Results to paste both the permutation and combination values into homework, notes, or spreadsheets.

The Logic Visualizer implements the counting principle. For P(6,2) without repetition, two slots show (6)×(5) = 30. For C(6,2), those 30 arrangements include duplicates (AB and BA); dividing by 2! = 2 gives 15 unique combinations. With repetition, e.g., P(10,3), slots show 10×10×10 = 1,000. The visualizer helps you see why permutations exceed combinations and how the factorial formulas arise from sequential choices, supporting both permutation formula and combination formula understanding.