How to Use
Select Permutation if order matters (arrangements), Combination if order doesn't matter (selections), or Factorial for n!.
Type the total number of items (n) and items chosen (r). Both must be non-negative integers with r ≤ n.
Click Calculate to see nPr or nCr with a full factorial breakdown and step-by-step working.
Frequently Asked Questions
Permutation counts arrangements where order matters (nPr). Combination counts selections where order does not matter (nCr). For example, choosing 2 from {A, B, C}: permutations = AB, BA, AC, CA, BC, CB = 6; combinations = AB, AC, BC = 3. nPr is always ≥ nCr.
nPr = n! / (n − r)! where n is the total items and r is items chosen. It expands as: nPr = n × (n−1) × (n−2) × … × (n−r+1). Example: 5P2 = 5 × 4 = 20.
nCr = n! / (r! × (n − r)!) = nPr / r!. Example: 5C2 = 5! / (2! × 3!) = 120 / (2 × 6) = 10. Also note: nCr = nC(n−r), so 10C7 = 10C3 = 120.
Use Permutation (order matters): arranging books on a shelf, forming 4-digit numbers, choosing president/VP/secretary from a group, assigning seats.
Use Combination (order doesn't matter): choosing a cricket team, selecting lottery numbers, forming a committee, picking toppings on a pizza.
0! = 1 by definition. This ensures formulas stay consistent — for example, nCn = n!/(n! × 0!) = 1, meaning there is exactly one way to choose all n items. The convention also follows from the pattern: 3! = 6, 2! = 3, 1! = 1, so 0! = 1.
nC0 = 1 (there is exactly one way to choose nothing) and nCn = 1 (there is exactly one way to choose all items). Also nC1 = n and nP1 = n — choosing or arranging 1 item from n gives n options.