Math
Permutations Calculator (nPr)
Calculate nPr, the number of ordered arrangements of r items chosen from n. Also shows nCr for comparison.
Permutations (nPr)
10P3 (order matters)
720
Number of ordered arrangements of 3 items chosen from 10.
For comparison: 10C3 (order does not matter)
120
Frequently Asked Questions about the Permutations Calculator (nPr)
What is a permutation?
A permutation is an ordered arrangement of items. nPr counts the distinct sequences of length r you can form from n items, where order matters and no item repeats. Choosing 3 books from a shelf of 10 and caring about which is first, second, or third is a permutation problem.
How is nPr different from nCr (combinations)?
Order matters in permutations; it does not in combinations. For the same n and r, nPr = nCr x r!. Assigning gold, silver, and bronze medals to 3 of 10 runners is nPr (10P3 = 720). Picking any 3 of those 10 runners for a relay team (rank ignored) is nCr (10C3 = 120).
What is the formula for nPr?
nPr = n! / (n - r)! = n x (n - 1) x ... x (n - r + 1). For 10P3: 10 x 9 x 8 = 720. The calculator also shows nCr = nPr / r!, so for that same example nCr = 720 / 6 = 120.
When should I use permutations in real life?
Any time order matters: ranking race finishers (1st, 2nd, 3rd), generating unique PIN sequences, scheduling the order of tasks, assigning distinct roles (president, vice president, treasurer) to club members, or counting lock-code sequences where 1-2-3 and 3-2-1 are different codes.
Why does the calculator switch to BigInt above n = 18?
JavaScript floating-point numbers can represent integers exactly only up to 2^53 - 1. Permutations with n >= 19 and r close to n exceed that limit, so the calculator switches to arbitrary-precision BigInt arithmetic. Results stay exact for any valid input up to n = 1000.