What is a combination in math?
A combination is a selection of items from a group in which the order does not matter. Picking Ana, Ben and Cy for a team is the same combination as Cy, Ana and Ben. The number of ways to choose r items from n is C(n, r) = n!r!(n − r)!; for example, C(5, 2) = 10.
Combination in a picture
Formal definition and notation
A combination of r objects from a set of n different objects, where 0 ≤ r ≤ n, is a subset of size r; its members have no order. The number of such subsets is C(n, r) = n!r!(n − r)!, also written nCr and read n choose r. Because each subset of r objects can be arranged in r! orders, C(n, r) equals the number of permutations P(n, r) divided by r!. Combinations count equally likely outcomes, so they are used to find probabilities such as the chance of a particular hand of cards.
| Symbol | Read it as | Meaning |
|---|---|---|
| C(n, r) | n choose r | the number of ways to choose r of n items; calculators label it nCr |
| n! | n factorial | n × (n − 1) × ⋯ × 2 × 1; for example 5! = 120, and 0! = 1 |
| P(n, r) | permutations of n taken r at a time | ordered arrangements, n!(n − r)!; always at least C(n, r) |
Key facts
- Ask first: does order matter? If yes, it is a permutation. If no, it is a combination.
- C(n, 0) = 1 and C(n, n) = 1: there is one way to choose nothing and one way to choose everything.
- C(n, 1) = n: choosing one item is the same as picking which one.
- Choosing what to take is the same as choosing what to leave: C(n, r) = C(n, n − r), so C(10, 8) = C(10, 2) = 45.
- C(n, r) = P(n, r)r!: for 2 of A, B, C, D, there are 12 ordered pairs but only 6 groups.
- In probability: P(event) = number of favorable combinations ÷ total combinations, when all choices are equally likely.
Combination formula
- n
- the number of items to choose from
- r
- the number of items chosen
- !
- factorial: multiply all whole numbers from that number down to 1
Shortcut: multiply r numbers counting down from n, then divide by r!. C(8, 3) = 8 × 7 × 63 × 2 × 1 = 3366 = 56.
Combination examples, step by step
Example 1: Pizza toppings
How many ways can you choose 3 toppings from a list of 7?
- Order does not matter: mushrooms, olives, peppers is the same pizza in any order.
- C(7, 3) = 7 × 6 × 53 × 2 × 1.
- = 2106 = 35.
Answer: There are 35 ways to choose the toppings.
Example 2: A committee with a condition
A club has 10 members. How many 4-person committees are possible? How many are possible if two particular members must be on the committee?
- Any 4 of 10: C(10, 4) = 10 × 9 × 8 × 74 × 3 × 2 × 1.
- 504024 = 210.
- If two members are already on it, choose the other 2 from the remaining 8 members.
- C(8, 2) = 8 × 72 × 1 = 28.
Answer: There are 210 possible committees, and 28 of them include both of those members.
Example 3: A probability with combinations
A bag holds 6 red and 4 blue marbles. Three marbles are picked at random at the same time. What is the probability that all three are red?
- All possible groups of 3 from 10 marbles: C(10, 3) = 120.
- Groups of 3 using only red marbles: C(6, 3) = 20.
- Probability: 20120 = 16.
Answer: The probability that all three are red is 16.
Common mistakes
210 counts every order of the same 3 toppings as different, but a pizza does not care about order.
Ana as president with Ben as vice president is a different result from Ben as president with Ana as vice president.
The formula divides by both r! and (n − r)!. Without (n − r)! you are not counting groups of 2 at all.
Not to be confused with
A permutation counts arrangements where order matters, such as gold, silver and bronze medals: P(5, 3) = 60. A combination counts groups where order does not matter: C(5, 3) = 10.
A factorial is one number, n! = n × (n − 1) × ⋯ × 1, such as 4! = 24. The combination formula is built from three factorials.
Despite its name, a lock code is not a combination in the math sense: 3-1-7 and 7-1-3 open different locks, so order matters, and the code is really a permutation.
Quick check: 3 questions
What is C(6, 2)?
Answer: B. C(6, 2) = 6 × 52 × 1 = 15. 30 is P(6, 2), which counts order; 12 is 6 × 2; 720 is 6!.Show answer and explanation
Which situation is counted with combinations?
Answer: D. Only the books are a plain group, where order does not matter. Rankings, PINs and different offices all depend on order, so they are permutations.Show answer and explanation
A pizza shop offers 9 toppings. How many different pizzas have exactly 2 different toppings?
Answer: C. C(9, 2) = 9 × 82 = 36. 72 counts each pair twice (in both orders), 18 is 9 × 2, and 81 is 9 × 9, which allows the same topping twice.Show answer and explanation
Combination: frequently asked questions
What is the difference between a permutation and a combination?
Order. In a permutation, changing the order gives a new result, as with race finishes or passwords. In a combination, only which items are chosen matters, as with a hand of cards or a team. That is why there are always at least as many permutations as combinations: P(5, 2) = 20, but C(5, 2) = 10.
How do you calculate nCr?
Use C(n, r) = n!r!(n − r)!. By hand, multiply r numbers counting down from n and divide by r!: C(9, 4) = 9 × 8 × 7 × 64 × 3 × 2 × 1, which is 302424 = 126. On a calculator, type 9, press nCr (often in the probability menu), then type 4.
Why is C(n, 0) equal to 1?
There is exactly one way to choose nothing: take no items. The formula agrees because 0! = 1, so C(n, 0) = n!0! × n! = 1. In the same way, there is exactly one way to choose all n items, so C(n, n) = 1 as well.
Practice and tools
Sources
- Common Core State Standards, High School Statistics and Probability, Conditional Probability and the Rules of Probability (HSS-CP.B.9)
- OpenStax, College Algebra 2e, 9.5 Counting Principles
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS HSS-CP.B.9
Cite this entry
Effortless Math Team. “Combination.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/combination/
Flashcard
Browse all terms in the Math Dictionary · More probability terms · Random term