combinatorics
Calculate n! = n × (n−1) × ... × 1.
This is the worked example that ships with the calculator. Replace the values above with your own and press Calculate.
Formulan! = n x (n-1) x (n-2) x ... x 1n = 5
120
Free reference
Combinatorics counts arrangements and selections. Factorials count orderings, permutations count ordered selections, and combinations count unordered selections.
n! = n * (n-1) * (n-2) * ... * 2 * 1 - The number of ways to arrange n distinct items in order, with 0! defined as 1.nPr = n! / (n - r)! - Ordered selections of r items from n distinct items.nCr = n! / (r! * (n - r)!) - Unordered selections of r items from n distinct items.A club with 10 members needs to choose a 3-person committee. The committee has no distinct roles, so order does not matter. How many different committees are possible?
Inputs. n = 10, r = 3.
nCr = n! / (r! * (n - r)!) gives n = 10, r = 3nCr = nP3 / 3! gives 120Answer: 120
There are 120 possible committees: 10! / (3! * 7!) = 720 / 6 = 120 unordered ways to pick 3 people from 10.
Permutations count ordered selections and combinations count unordered ones. Choosing a president, secretary and treasurer from 10 people is a permutation (10P3 = 720); choosing a 3-person committee is a combination (10C3 = 120).
n! is the product of every whole number from n down to 1, so 5! = 5 * 4 * 3 * 2 * 1 = 120. It counts the number of ways to arrange n distinct items, and 0! is defined to be 1.
Every unordered selection of r items can be arranged in r! different orders, so the permutation count is exactly r! times the combination count. Dividing nPr by r! removes the orderings, which is why 10C3 = 720 / 6 = 120.