Examples of Combinations and Permutations
A permutation is an arrangement where order matters; a combination is a selection where order does not matter. Enter n and r to compare both counts and see the exact working.
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Ranking three winners is a permutation. Choosing three committee members is a combination.
nCr and nPr example calculator
Use whole numbers with 0 ≤ r ≤ n. Results are exact integers, including large counts through n = 500.
Permutation and combination formulas
| Situation | Formula | Order |
|---|---|---|
| Permutation without repetition | nPr = n! ÷ (n − r)! | Matters |
| Combination without repetition | nCr = n! ÷ [r!(n − r)!] | Does not matter |
| Ordered choices with repetition | nr | Matters |
| Combinations with repetition | (n + r − 1)Cr | Does not matter |
The first two formulas are related by nPr = nCr × r!. Every unordered selection has r! possible internal orders.
Worked combination and permutation examples
Example 1: 5C2 combination
Choosing 2 students from 5 is a combination because choosing Ava then Ben is the same pair as choosing Ben then Ava. Therefore 5C2 = 5! ÷ (2!3!) = 10.
Example 2: award three places
Awarding gold, silver, and bronze among 6 finalists is ordered. Use 6P3 = 6 × 5 × 4 = 120. Choosing three finalists without assigning places would instead be 6C3 = 20.
Example 3: nPr in probability
If one of 9 horses must finish first and another second, there are 9P2 = 72 ordered results. One specified exact result therefore has probability 1/72 when all ordered outcomes are equally likely.
How many possible combinations of 4 numbers?
The question needs a selection size and a repetition rule. Choosing all 4 distinct numbers as one unordered group gives 4C4 = 1. Arranging all four gives 4! = 24 permutations. Choosing 2 from the four gives 4C2 = 6 combinations. State whether order and repetition matter before calculating.
What changes when repetition is allowed?
Without repetition, an item can appear only once in a result. With repetition, a four-digit code made from ten digits has 104 = 10,000 ordered outcomes. For a visible list rather than only the count, use the permutation generator.
Frequently asked questions
What is the fastest way to tell nCr from nPr?
Ask whether swapping the selected objects creates a new outcome. If yes, use a permutation; if no, use a combination.
What does 5C2 mean?
It means the number of unordered selections of two objects from five. The answer is 10.
Can r be greater than n?
Not when selecting without replacement. The calculator requires 0 ≤ r ≤ n.