worldofcalcsPermutation and Combination Calculator
Permutations, nPr
Order matters
720
10P3 = 10 × 9 × 8 = 720
Combinations, nCr
Order doesn't matter
120
10C3 = 10P3 ÷ 3! = 720 ÷ 6 = 120
With repetition allowed
Arrangements, nʳ
Order matters
1,000
10³
Selections, (n + r − 1)Cr
Order doesn't matter
220
12C3
What are permutations and combinations?
Both count the ways to choose r items from n. In a permutation the order matters, as with first, second and third prizes or a PIN. In a combination it doesn't, as with picking a team or a set of questions to answer. Because each group of r can be arranged in r! orders, there are always r! times as many permutations as combinations.
nPr and nCr formulas
nPr = n! ÷ (n − r)! · nCr = n! ÷ (r! × (n − r)!) = nPr ÷ r! · With repetition: nʳ arrangements, (n + r − 1)Cr selections
n! (n factorial) is n × (n − 1) × … × 1, and 0! = 1. With repetition allowed, the same item can be chosen more than once, like the digits of a PIN.
Permutation or combination?
- Ask whether swapping two chosen items gives a different result. If it does (ranks, seats, passwords), use permutations; if not (teams, hands of cards), use combinations.
- nCr = nC(n − r): choosing 3 to include from 10 is the same as choosing 7 to leave out, so 10C3 = 10C7 = 120.
- nC0 = nCn = 1 and nC1 = n.
- Arranging all n items in a row gives n! ways: 5 books can be shelved in 5! = 120 orders.
- A 5-card hand from a 52-card deck can be any of 52C5 = 25,98,960 combinations.
Worked example: choosing 3 people from 10
- A club has 10 members and needs a president, a secretary and a treasurer
- Order matters (the roles differ): 10P3 = 10 × 9 × 8 = 720 ways
- For a committee of 3 with no roles, order doesn't matter
- 10C3 = 10P3 ÷ 3! = 720 ÷ 6 = 120 ways
- A 3-digit code from the digits 0–9, repeats allowed: 10³ = 1,000 codes
Common nPr and nCr values
Values that come up in textbook and exam questions.
| n | r | nPr | nCr |
|---|---|---|---|
| 4 | 2 | 12 | 6 |
| 5 | 3 | 60 | 10 |
| 6 | 2 | 30 | 15 |
| 8 | 3 | 336 | 56 |
| 10 | 3 | 720 | 120 |
| 11 | 5 | 55,440 | 462 |
| 12 | 4 | 11,880 | 495 |
| 52 | 5 | 31,18,75,200 | 25,98,960 |
nPr is always nCr × r!.