Combination Calculator

Calculate combinations nCr, the number of ways to choose r items from n when order does not matter, with or without repetition. Shows the steps, the exact number for large inputs and the odds of one specific selection. Example: 52C5 = 2,598,960 poker hands.

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Total items (n)

Items chosen (r)

Repetition

How to use the Combination Calculator

Enter the total number of items n and how many you are choosing, r.
Choose whether the same item may be picked more than once.
Press Calculate for the exact count and the working.

Formula and method

nCr = n! / (r! (n – r)!). With repetition allowed: C(n + r – 1, r). Symmetry: nCr = nC(n – r). Pascal’s rule: nCr = (n-1)C(r-1) + (n-1)Cr.

Worked examples

Question	Calculation	Answer
Poker hands from 52 cards	52C5	2,598,960
Committee of 3 from 10 people	10C3 = 720 / 6	120
Pick 6 of 49 lottery numbers	49C6	13,983,816
Scoops of 3 from 4 flavors, repeats allowed	C(6, 3)	20

Combination versus permutation

	Combination	Permutation
Order matters?	No	Yes
10 and 3	120	720
Relationship	nPr / r!	nCr x r!

Pascal’s triangle row values

n	C(n,0)	C(n,1)	C(n,2)	C(n,3)	C(n,4)
4	1	4	6	4	1
5	1	5	10	10	5
6	1	6	15	20	15

Frequently asked questions

What is a combination?

A selection of items where order does not matter.

What is the formula for nCr?

n! divided by r! times (n – r)!.

What is 5 choose 2?

10.

How many 5-card hands are there?

2,598,960.

Is nCr the same as nC(n-r)?

Yes, choosing r to include is the same as choosing n – r to leave out.

What does repetition allowed mean?

The same item may be chosen more than once, as when picking ice cream scoops.

Sources

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Effortless Math Team

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