Permutations and combinations

Permutations count arrangements where order matters; combinations count selections where it does not. Both count ways to pick r items from n without repetition.

ByHoang TruongUpdated July 5, 2026

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A firm shortlists 8 candidates for three distinct roles, CEO, CFO and COO. Because order matters, swapping two people changes the outcome, this is a permutation: P(8,3) = 8 × 7 × 6 = 336 ways. Choosing the same 8 candidates for an unordered 3-person advisory panel instead is a combination: C(8,3) = 336 ÷ 3! = 336 ÷ 6 = 56 ways.

Where it fits

TopicProbability & DistributionsCoreSubjectData Analysis & StatisticsCore

The formula

LaTeX

{}_{n}P_{r} = \frac{n!}{(n-r)!}

Variables

$n$: Total number of distinct items available
$r$: Number of items being arranged or selected

Number of ordered arrangements of r items chosen from n distinct items, without repetition.

LaTeX

{}_{n}C_{r} = \frac{n!}{r!(n-r)!}

Variables

$n$: Total number of distinct items available
$r$: Number of items being selected

Number of unordered selections of r items chosen from n distinct items, without repetition; equals nPr divided by r!.

Check yourself

PracticeCORE

A company has shortlisted 10 suppliers and must award contracts for 4 distinct regions: North, South, East and West. No supplier can hold more than one regional contract. How many different ways can the regional contracts be awarded?

Select an answer to check your understanding.