🎲 Permutations & Combinations Calculator

Compute nPr (permutations — order matters) and nCr (combinations — order doesn't matter) for any n and r.

n (total items)

r (chosen items)

One counts arrangements, the other counts groups

A permutation counts ordered picks; a combination divides that by the number of ways the same r items could have been ordered:

nPr = n! / (n − r)! = n × (n−1) × ... × (n−r+1)
nCr = nPr / r! = n! / (r! × (n − r)!)

With the defaults n = 10 and r = 3, nPr is 10 × 9 × 8 = 720 and nCr is 720 / 6 = 120. The card also prints 10! = 3,628,800. The tool uses the falling product on the right, multiplying three terms rather than building 10! and dividing.

Choosing between them, and where the limits are

Order matters, use nPr. Gold, silver and bronze from eight runners is 8 × 7 × 6 = 336; just knowing who medalled is 56.
Lottery odds are combinations. Six numbers from 49 gives 13,983,816 tickets.
nCr is symmetric: choosing 3 from 10 and choosing 7 from 10 both give 120, because every group you pick defines the group you left.
r = 0 returns 1 either way - one way to pick nothing. An r above n is rejected, and so is n above 170, where the factorial overflows to infinity.

Neither figure allows repeats. A four-digit PIN, where each digit can recur, is 10⁴ = 10,000, not a permutation. Factorials past 18! also exceed exact whole-number precision, so very large answers are close rather than exact.

Frequently asked questions

What is 10 choose 3?

120. There are 720 ordered ways to pick three from ten, and each set of three can be ordered 6 ways.

What is the difference between a permutation and a combination?

Order. ABC and CBA are two permutations but one combination, which is why nPr is always the larger of the two whenever r is 2 or more.

How many combinations are there in a 6 from 49 lottery?

13,983,816. Enter n = 49 and r = 6; the nCr row is your odds of one line matching all six.