Combination Calculator
Enter n, the number of items, and r, how many you choose, to count the unordered selections nCr. Switch on repetition for cases such as ice cream scoops.
Last updated November 3, 2026
Result
C(10, 3) = 120
| Combinations (order ignored) | 120 |
|---|---|
| Permutations for comparison | 720 |
| n! | 3,628,800 |
| Digits in the answer | 3 |
Show the work
Formula: C(n, r) = n! / (r! (n - r)!)
- n = 10, r = 3, repetition not allowed.
- C(10, 3) = P(10, 3) / 3! = 720 / 6.
- C(10, 3) = 120.
Result: C(10, 3) = 120
How combinations are counted
C(n, r) = n! ÷ (r! × (n − r)!)
A combination is a selection in which order does not matter. Picking Ann and Ben for a committee is the same group as picking Ben and Ann. To count them, take the ordered arrangements, P(n, r), and divide by r!, the number of ways to reorder each chosen group.
This gives C(n, r) = n! ÷ (r! × (n − r)!), also written nCr or “n choose r”. For 10 items taken 3 at a time it is 720 ÷ 6 = 120. The result is symmetric, so choosing 3 to include is the same as choosing 7 to leave out: C(10, 3) = C(10, 7).
When items can be chosen more than once, such as scoops of ice cream, the count is C(n + r − 1, r). The calculator returns exact whole numbers, so even large results keep every digit.
See how we calculate for the formulas, sources and review schedule behind every calculator.
Also useful alongside this calculator: Probability Calculator, Statistics Calculator and Sample Size Calculator.
When order does not matter
Use combinations for committees, hands of cards, lottery tickets and pizza toppings. A club choosing a committee of 4 from 12 members has C(12, 4) = 495 possible committees. Choosing 3 toppings from 8 gives C(8, 3) = 56 pizzas.
A five-card poker hand is a combination of 5 cards from 52, so there are C(52, 5) = 2,598,960 hands. A 6-number lottery from 49 balls has C(49, 6) = 13,983,816 tickets, so a single ticket wins with probability 1 in 13,983,816.
Choosing with repetition
Three scoops from 5 flavors, where a flavor can be repeated and the order of scoops does not matter, can be chosen in C(5 + 3 − 1, 3) = C(7, 3) = 35 ways. Here r can be larger than n, because a flavor can be used again.
Without repetition r cannot exceed n. The calculator flags that case and suggests switching the repetition option if items can be reused.
Pascal’s triangle and useful identities
Combinations are the entries of Pascal’s triangle. Each entry is the sum of the two above it: C(5, 2) = C(4, 1) + C(4, 2) = 4 + 6 = 10. A full row adds up to a power of two, since C(5, 0) through C(5, 5) are 1, 5, 10, 10, 5 and 1, which total 32.
The number of handshakes at a party where all 10 guests shake hands once is C(10, 2) = 45, because each handshake is an unordered pair of people.
Very large results
The counts grow quickly, and ordinary floating-point arithmetic loses digits beyond about 15 places. This calculator multiplies and divides exact integers, so every digit is correct. Results over 24 digits are shown in scientific notation, with the digit count in the table.
The limit is 1,000 for both n and r. For ordered arrangements of the same items, use the Permutation Calculator.
How to use the Combination Calculator
- Enter n, the total number of distinct items.
- Enter r, how many of them you choose, and set whether items can repeat.
- Read nCr, with the permutation count shown for comparison.
- Open Show the work for the division, then use Copy, Print or Share.
Worked example
A teacher picks 3 students from 10 for a team. The ordered picks number P(10, 3) = 720, and each team of three appears in 3! = 6 orders, so there are 720 ÷ 6 = 120 different teams. That is C(10, 3) = 120.
| n | r | What it counts | Combinations |
|---|---|---|---|
| 5 | 2 | Pairs from 5 people | 10 |
| 8 | 3 | Pizza toppings | 56 |
| 12 | 4 | Committees | 495 |
| 52 | 5 | Poker hands | 2,598,960 |
| 49 | 6 | Lottery tickets | 13,983,816 |
Frequently asked questions
What is the difference between a combination and a permutation?
A combination ignores order and a permutation counts it. For the same n and r, there are r! times as many permutations as combinations.
What does nCr mean?
nCr, also written C(n, r) or “n choose r”, is the number of ways to choose r items from n distinct items when the order of selection does not matter.
What is C(n, 0) and C(n, n)?
Both equal 1. There is exactly one way to choose nothing and exactly one way to choose every item.
Can r be larger than n?
Only when repetition is allowed. Without repetition you cannot choose more distinct items than exist, so the calculator asks you to fix r.
Are my numbers stored or sent anywhere?
No. The calculation runs in your browser and nothing you enter is saved or uploaded unless you copy or share a link yourself.