What is the difference between a combination and a permutation?

A combination counts the number of ways to choose r items from n items when order does not matter. A permutation counts the number of ways to arrange r items from n when order does matter. For example, choosing 3 players from a squad of 10 to start a game is a combination (C(10,3) = 120) because the order in which you pick them makes no difference to the result. Lining those same 3 players up in batting order is a permutation (P(10,3) = 720) because the first, second, and third positions are distinct. The relationship is nCr = nPr / r!.

How do you calculate nCr by hand?

Use the formula C(n,r) = n! / (r! x (n-r)!). For C(10,3): the numerator is 10! = 3,628,800; the denominator is 3! x 7! = 6 x 5,040 = 30,240. Dividing gives 120. In practice you can simplify by cancelling the larger factorial first: C(10,3) = (10 x 9 x 8) / (3 x 2 x 1) = 720 / 6 = 120. Always start from the top with r terms and divide by r! to keep the numbers manageable.

Why does C(n,r) equal C(n, n-r)?

Choosing r items to include is mathematically identical to choosing (n-r) items to exclude. For example, C(10,3) = 120 and C(10,7) = 120 because picking 3 items out of 10 leaves 7, and picking which 7 to leave out is the same choice. This symmetry is expressed by the formula because swapping r and (n-r) in n! / (r! x (n-r)!) gives the same denominator. It is also visible in Pascal's Triangle, where each row is symmetric.

Combination (nCr) Calculator

Calculate C(n,r) instantly: the number of ways to choose r items from a set of n items when order does not matter. Enter any whole numbers for n and r and the result updates immediately with full step-by-step working.

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Standard formula C(n,r) = n! / (r! x (n-r)!) as defined in combinatorics.

1. Inputs

n (total number of items)

r (items to choose)

2. Quick Reference

Load a common example

Result

C(n, r)
120
Number of combinations

nPr (for comparison)

720

Permutations (order matters)

r! (divisor)

nCr = nPr / r!

Step-by-step working

Calculation Details

n (total items)10

r (items chosen)3

n - r7

n!3,628,800

r!6

(n-r)!5,040

r! x (n-r)!30,240

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

Symmetry Check

C(n, r)120

C(n, n-r)120

These are equal?Yes

Total arrangements nPr720

nPr / r! = nCr720 / 6 = 120

n choose 01

n choose n1

What is a Combination?

A combination is a selection of items from a larger set where order does not matter. If you are choosing a 3-person committee from 10 candidates, it does not matter whether you pick Alice first or last -- the committee is the same. C(10,3) = 120 tells you there are 120 distinct committees possible.

Combinations are written as C(n,r), nCr, or ⁿC_r. The notation "n choose r" is also standard. In Pascal's Triangle, C(n,r) appears as the (r+1)th element of the (n+1)th row.

The Combination Formula

The combination formula is:

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

Where n! ("n factorial") means n x (n-1) x (n-2) x ... x 1. For example, 5! = 5 x 4 x 3 x 2 x 1 = 120.

A practical shortcut: instead of computing full factorials, write out only r terms in the numerator starting from n, then divide by r!. For C(10,3):

(10 x 9 x 8) / (3 x 2 x 1) = 720 / 6 = 120

Worked Example: Default Values

With n = 10 and r = 3:

Numerator: 10 x 9 x 8 = 720
Denominator: 3! = 3 x 2 x 1 = 6
C(10,3) = 720 / 6 = 120

There are 120 ways to choose any 3 items from a set of 10 when the order of selection does not matter.

Combinations vs Permutations

Concept	Formula	C(10,3) example	When to use
Combination (nCr)	n! / (r! x (n-r)!)	120	Order does not matter (teams, committees, hands)
Permutation (nPr)	n! / (n-r)!	720	Order matters (rankings, arrangements, sequences)
Relationship	nCr = nPr / r!	720 / 6 = 120	Divide permutations by the number of orderings of r items

Common Applications

Lottery odds: New Zealand Lotto draws 6 numbers from 40. There are C(40,6) = 3,838,380 possible combinations.
Poker: A 5-card hand from a 52-card deck has C(52,5) = 2,598,960 possible hands.
Teams and committees: Selecting 4 people from a group of 20 gives C(20,4) = 4,845 possible teams.
Probability: The probability of a specific combination is 1 / C(n,r) when each outcome is equally likely.
Binomial theorem: The binomial coefficient C(n,r) is the coefficient of x^r in the expansion of (1 + x)ⁿ.

Special Cases

C(n,0) = 1: There is exactly one way to choose nothing.
C(n,1) = n: Choosing one item from n gives n possibilities.
C(n,n) = 1: There is exactly one way to choose all items.
C(n,r) = C(n, n-r): Choosing r to include is the same as choosing (n-r) to exclude.

Related Calculators

Maths and Stats Calculators: full hub for maths tools.
Permutation Calculator (nPr): ordered arrangements from n items.
Permutations and Combinations Calculator: both in one tool.
Factorial Calculator: compute n! for any whole number.
Probability Calculator: use combinations in probability problems.

Method: Standard combinatorics formula C(n,r) = n! / (r! x (n-r)!). Factorials are computed iteratively using JavaScript BigInt for exact integer arithmetic at any practical value of n. The calculator handles values up to n = 170 before results exceed JavaScript's safe integer range; for larger values BigInt is used throughout.

This calculator computes exact integer results using the standard combination formula. Results are exact for all inputs shown. For very large n (above a few hundred), the number of digits in the result grows rapidly.

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