What is the formula for combinations without repetition?

The formula is C(n, r) = n! / (r! x (n - r)!), where n is the total number of items to choose from and r is the number of items you select. The exclamation mark means factorial, so 5! = 5 x 4 x 3 x 2 x 1 = 120. This formula counts how many unique groups of r items can be formed from n distinct items when order does not matter and each item can only be selected once.

What is the difference between combinations with and without repetition?

Combinations without repetition (this calculator) means each item from the total set can appear at most once in any selection. For example, choosing 3 cards from a standard deck without putting any back. The formula is C(n, r) = n! / (r! x (n - r)!). Combinations with repetition allow an item to be chosen more than once. For example, choosing 3 scoops of ice cream from 5 flavours where you can pick the same flavour multiple times. That formula is C(n + r - 1, r) = (n + r - 1)! / (r! x (n - 1)!).

How does a combination differ from a permutation?

A combination counts selections where order does not matter, while a permutation counts arrangements where order does matter. For example, choosing the team {Alice, Bob, Carol} is the same combination regardless of which name you list first. But the arrangement Alice-Bob-Carol is a different permutation from Bob-Alice-Carol. For any given combination of r items from n, there are r! corresponding permutations. So the number of combinations C(n, r) always equals the number of permutations P(n, r) divided by r!.

Combination Without Repetition Calculator

This calculator works out how many ways you can choose r items from a set of n distinct items when order does not matter and each item can only be chosen once, using the standard combination formula C(n, r) from probability, statistics and combinatorics. Enter the total number of items (n) and the number you want to select (r), or pick a quick preset such as a poker hand, a Lotto draw or a committee selection. The calculator returns the number of combinations C(n, r), the permutations P(n, r) for the same n and r, and r factorial, the order factor linking the two, along with step-by-step working showing n factorial, r factorial, (n minus r) factorial and the final division. It also shows an interpretation panel with the total selections possible, the ordered arrangements, the odds of one specific combination occurring, and that probability as a percentage, plus a plain English summary of the result. Use it whenever you need to know how many unique groups or hands are possible regardless of order, such as lottery odds, poker hands, sports fixtures or committee line-ups. Because factorials grow extremely quickly, results for large n are shown in scientific notation once they exceed JavaScript's safe integer range, though the calculation stays exact up to n = 170.

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Standard Formula C(n, r) = n! / (r! x (n - r)!). Standard combinatorics formula, verified against textbook examples.

Combinations Without Repetition

C(n, r) = n! / (r! × (n − r)!)

n = total items | r = items chosen | ! = factorial

Inputs

Total items (n) How many items in the set

Items chosen (r) How many you select

Quick Presets

Load a common example

Please enter valid whole numbers where n ≥ r ≥ 0 and n ≥ 1.

Result

C(n, r)
120
Combinations without repetition

Permutations P(n, r)

720

Ordered arrangements

r! (order factor)

C(n,r) × r! = P(n,r)

Step-by-Step Working

n (total items)10

r (items chosen)3

n - r7

n!3,628,800

r!6

(n - r)!5,040

C(n, r) = n! / (r! x (n-r)!)120

Interpretation

Total selections possible120

Ordered arrangements P(n,r)720

Order divides result by r!6

Probability of 1 specific combo1 in 120

As a percentage0.833%

Summary: There are 120 ways to choose 3 items from 10 distinct items when order does not matter and no item is repeated.

What is a Combination Without Repetition?

A combination without repetition counts the number of ways to select a group of r items from a set of n distinct items where order does not matter and each item can only appear once in a selection. This is the most common type of combination used in everyday probability, statistics, and exam questions.

For example, if you are choosing a committee of 3 people from a group of 10, the committee {Alice, Bob, Carol} is the same as {Carol, Alice, Bob} - the order you list them does not matter, and the same person cannot appear twice. This is a combination without repetition.

The Formula

The number of combinations without repetition is written C(n, r), also written as nCr or "n choose r":

C(n, r) = n! / (r! × (n − r)!)

Where n! means n factorial: the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. By convention, 0! = 1.

Worked Example: Default Values (n=10, r=3)

Choosing 3 items from 10 distinct items:

Step	Calculation	Value
Calculate n!	10!	3,628,800
Calculate r!	3!	6
Calculate (n - r)!	7!	5,040
Denominator	3! × 7!	30,240
C(10, 3)	3,628,800 / 30,240	120

There are 120 unique ways to choose 3 items from a set of 10 when order does not matter and each item is only chosen once.

Combinations vs Permutations

Permutations count ordered arrangements: P(n, r) = n! / (n - r)!. Because there are r! ways to arrange any group of r items, permutations always equal combinations multiplied by r!.

Scenario	Order Matters?	Repetition?	Formula
Combinations without repetition	No	No	n! / (r! × (n-r)!)
Combinations with repetition	No	Yes	(n+r-1)! / (r! × (n-1)!)
Permutations without repetition	Yes	No	n! / (n-r)!
Permutations with repetition	Yes	Yes	n^r

Real-World Examples

Lotto (NZ): Choosing 6 numbers from 40 gives C(40, 6) = 3,838,380 combinations.
Poker hand: A five-card hand from a 52-card deck gives C(52, 5) = 2,598,960 unique hands.
Committee selection: Choosing a 3-person committee from 10 candidates gives C(10, 3) = 120 possible committees.
Sports fixtures: If 8 teams each play every other team once, there are C(8, 2) = 28 matches in a round-robin.

Boundary Cases

C(n, 0) = 1 for any n: there is exactly one way to choose nothing.
C(n, n) = 1: there is exactly one way to choose all items.
C(n, 1) = n: choosing a single item from n gives n options.
C(n, r) = C(n, n - r): choosing r items to include is the same count as choosing n - r items to exclude.

Related Calculators

Maths and Stats Calculators: browse the full maths and stats collection.
Combination nCr Calculator: another interface for the same C(n,r) formula.
Permutation nPr Calculator: count ordered arrangements with P(n,r).
Combinations with Repetition Calculator: when items can be chosen more than once.
Probability Calculator: use combination results to work out event probabilities.
Factorial Calculator: compute n! for large values.

Method: Standard combinatorics formula C(n, r) = n! / (r! × (n - r)!). Factorials are computed iteratively to avoid floating-point rounding for values up to n = 170 (JavaScript's safe range). Results above this range are shown in scientific notation.

This calculator computes exact integer results for n up to approximately 170. For very large values, JavaScript floating-point arithmetic may cause slight rounding in the final digits. The formula and worked examples have been independently verified against standard combinatorics textbooks.