Probability Updated November 2026

Permutations and combinations

A permutation counts arrangements where order matters: nPr = n! / (n - r)!. A combination counts selections where order does not matter: nCr = n! / (r! * (n - r)!). Picking a president and treasurer is a permutation; picking a committee of two is a combination.

Syntax

nPr = n! / (n - r)!;   nCr = n! / (r! * (n - r)!)

Part	What it is
`n`	the number of items available
`r`	how many you pick
`n!`	n factorial, n * (n - 1) * ... * 1, with 0! = 1

Examples

Permutations of 2 from 5 (gold and silver among 5 runners)
5P2 = 5! / 3! = 5 * 4

Result: 20

Excel: =PERMUT(5, 2)

Combinations of 2 from 5 (any two of 5 toppings)
5C2 = 5! / (2! * 3!) = 20 / 2

Result: 10

Excel: =COMBIN(5, 2)

10C3 = 10 * 9 * 8 / (3 * 2 * 1)

Result: 120

Lottery: choose 6 of 49, order irrelevant
49C6

Result: 13,983,816

Common mistakes

Using nPr when order does not matter; it overcounts by a factor of r!, so divide by r! or use nCr.
Treating repeated items or repeated picks with these formulas; picks with replacement use n^r, and words with repeated letters need extra division.
Computing 0! as 0; it is 1, which is why nC0 = 1 and nCn = 1.

Practice Permutations and combinations

These drills check your answer on live data, free, no account needed.

Permutations: ordered choicesThere are 6 speakers and 2 distinct time slots in a conference session to fill. No one fills two of them. How many different ordered arrangements are possible? Give a whole number.
Combinations: unordered choicesAn analyst must pick 3 of 8 survey questions to keep in a short form. The order of picking does not matter. How many different groups are possible? Give a whole number.
Sampling without replacementA shipment of 10 laptops contains 3 with a faulty battery. An inspector picks 5 laptops at random, without replacement. What is the probability that exactly 2 of them have a faulty battery? Round to 4 decimal places.

Questions

How do I know whether to use permutations or combinations?

Ask whether swapping two chosen items gives a different outcome. If yes (rankings, passwords, seating) use permutations. If no (teams, hands of cards, lottery numbers) use combinations.

What is nCr called?

The binomial coefficient, read as n choose r. It appears in the binomial distribution and in Pascal's triangle.

How many ways can 5 people line up?

5! = 120. Arranging all n items is nPn = n!.

Probability rulesThe basic probability rules with dice and coin examples: the addition rule for A or B, the multiplication rule for independent events and the complement rule.
Binomial distributionThe binomial formula P(X = k) = nCk p^k (1 - p)^(n - k), its mean np and standard deviation sqrt(np(1 - p)), exact worked examples and Excel BINOM.DIST.
Expected valueExpected value is the probability-weighted average of all outcomes. Formula, worked examples with dice, raffles and bets, and the Excel SUMPRODUCT formula.
Conditional probabilityHow to calculate conditional probability P(A|B) = P(A and B) / P(B), read it from a two-way table, check independence and avoid confusing P(A|B) with P(B|A).