Combination Calculator
Find the number of combinations (nCr) and permutations (nPr) when choosing r items from n, with and without repetition.
Results
| Item | Value |
|---|
Steps
| Step | Working |
|---|
Quick Answer
Combinations count groups where order does not matter: nCr = n! ÷ (r! × (n − r)!). Choosing 3 people from 10 gives 120 combinations. Permutations count ordered arrangements: nPr = n! ÷ (n − r)!, so there are 720 ways to award gold, silver and bronze among 10 people.
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Table of Contents
- Combination Calculator
- Quick Answer
- Related Calculators
- How This Combination Calculator Works
- How to Use the Combination Calculator
- Core Formulas
- Key Terms
- Worked Example
- Real-Life Examples
- Combination vs Permutation
- Combinations for Small n
- Choosing the Right Formula
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Combination Calculator Works
Enter the total number of items (n) and how many you choose (r). The calculator gives four answers: combinations and permutations, each with and without repetition. It uses exact whole numbers, so even large answers are precise.
Quick explanation: combinations ignore order, permutations count it.
- Combination: a team of 3 from 10 people. ABC is the same as CBA.
- Permutation: 1st, 2nd and 3rd place. ABC is different from CBA.
- Repetition: items can be picked more than once, like digits in a PIN.
How to Use the Combination Calculator
- Enter n, the total number of items to choose from.
- Enter r, how many items you choose.
- Press Calculate.
- Read the combinations and permutations, with and without repetition.
- Check the steps table to see how nCr and nPr were worked out.
Core Formulas
- Combinations: nCr = n! ÷ (r! × (n − r)!)
- Permutations: nPr = n! ÷ (n − r)!
- Combinations with repetition: (n + r − 1)! ÷ (r! × (n − 1)!)
- Permutations with repetition: nʳ
n! (n factorial) means n × (n − 1) × … × 1, and 0! = 1.
Key Terms
- Factorial (n!): the product of all whole numbers from 1 to n.
- Combination: a selection of items where order does not matter.
- Permutation: an arrangement of items where order matters.
- Binomial coefficient: another name for nCr, written as n over r in brackets.
Worked Example
How many 5-card hands can be dealt from a 52-card deck?
- Order does not matter, so use combinations
- 52C5 = 52! ÷ (5! × 47!)
- = (52 × 51 × 50 × 49 × 48) ÷ 120 = 2,598,960 hands
Real-Life Examples
- A lottery: picking 6 numbers from 49 gives 13,983,816 combinations, so one ticket has a 1 in 13,983,816 chance.
- A 4-digit PIN: digits 0-9 can repeat and order matters, so there are 10⁴ = 10,000 PINs.
- Pizza toppings: choosing 3 toppings from 12 gives 220 different pizzas.
- A race: the first three places from 8 runners can finish in 336 different orders.
Combination vs Permutation
- Combination: a selection where order does not matter, such as a committee.
- Permutation: an arrangement where order matters, such as a race result or password.
- The link: nPr = nCr × r!, because each group of r can be arranged r! ways.
- Memory aid: “permutation” and “position” both start with P.
Combinations for Small n
| n | r = 2 | r = 3 | r = 4 | r = 5 |
|---|---|---|---|---|
| 5 | 10 | 10 | 5 | 1 |
| 6 | 15 | 20 | 15 | 6 |
| 8 | 28 | 56 | 70 | 56 |
| 10 | 45 | 120 | 210 | 252 |
| 12 | 66 | 220 | 495 | 792 |
| 15 | 105 | 455 | 1,365 | 3,003 |
| 20 | 190 | 1,140 | 4,845 | 15,504 |
Choosing the Right Formula
- Does order matter? If yes, use permutations. If no, use combinations.
- Can items repeat? If yes, use the repetition versions.
- Locks and PINs: order matters and digits repeat, so use nʳ.
- Lottery tickets: order usually does not matter and numbers do not repeat, so use nCr.
Common Mistakes to Avoid
- Using permutations when order does not matter. That overcounts by r!.
- Forgetting 0! = 1. It is defined as 1, so nC0 and nCn both equal 1.
- Choosing more than you have. Without repetition, r cannot be larger than n.
- Ignoring repetition. PINs and dice rolls allow repeats.
- Calculating huge factorials by hand. Cancel terms first, or let the calculator do it.
Frequently Asked Questions
Use nCr = n! ÷ (r! × (n − r)!). For 10 choose 3, that is 3,628,800 ÷ (6 × 5,040) = 120.
nCr ignores order and nPr counts it. nPr is always nCr × r!.
With digits 0-9 that can repeat, there are 10,000.
0! = 1, by definition. This keeps the formulas working when r is 0 or equal to n.
It equals 1, because there is exactly one way to choose nothing.
No. Order matters on a lock, so mathematically it is a permutation.
5! = 120 ways.
52C5 = 2,598,960.
Divide the number of favorable combinations by the total combinations. The chance of being dealt a specific 5-card hand is 1 in 2,598,960.
8C2 = 28 different pairs, because order does not matter within a pair.
Each item can be picked again, so there are more possible groups. Choosing 3 scoops from 10 flavors, with repeats allowed, gives 220 options instead of 120.
A triangle of numbers where each one is the sum of the two above it. Row n lists all the values of nCr.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Combination | Combination formulas. | https://en.wikipedia.org/wiki/Combination |
| Wikipedia, Permutation | Permutation formulas. | https://en.wikipedia.org/wiki/Permutation |