Calculate nPr (permutations) and nCr (combinations) from n and r — with factorials shown and the full formula applied step by step.
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Enter n (total items) and r (items chosen) — see both nPr and nCr with factorials and full formulas.
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Permutations count ordered arrangements — the order of selection matters. Combinations count unordered selections — the order does not matter. Choosing 2 letters from {A,B,C}: permutations = AB, BA, AC, CA, BC, CB (6 options); combinations = AB, AC, BC (3 options). Key question: does swapping two items give a different outcome?
nPr = n! / (n−r)! — permutations of n items taken r at a time. nCr = n! / (r! × (n−r)!) — combinations. Relationship: nPr = nCr × r!, because each combination can be arranged in r! different orders. For n=10, r=3: 10P3 = 720 and 10C3 = 120.
n! (n factorial) is the product of all positive integers from 1 to n. 5! = 5×4×3×2×1 = 120. By definition 0! = 1. Factorials grow extremely rapidly: 20! ≈ 2.4×10¹⁸. They appear throughout combinatorics, probability, the binomial theorem and Taylor series.
Use permutations when order matters: PIN codes (1234 ≠ 4321), race podium placements, seating arrangements, password generation. Use combinations when order is irrelevant: choosing a committee, lottery numbers, card hands, selecting toppings on a pizza. Ask yourself: "Does changing the order give a different result?"
Every number in Pascal\ Triangle equals C(n,r) where n is the row and r is the position (both starting at 0). Row 4 reads: 1 4 6 4 1 = C(4,0) C(4,1) C(4,2) C(4,3) C(4,4). Pascal\ Triangle shows all combinations at a glance and directly gives the binomial expansion coefficients.
