Combination Calculator (nCr)
Calculate combinations nCr — the number of ways to choose r items from n when order does not matter. Free combination calculator with and without repetition.
Updated 2026-06-14 · Free · No sign-up · Runs privately in your browser
Show the formula & steps
How the Combination Calculator Works
Enter the total number of items n and how many you are choosing r, and this calculator returns the number of combinations (nCr) — selections where order does not matter — plus the count with repetition. It is computed iteratively to stay accurate for large values like lottery odds.
The Formula
nCr = n! ÷ (r! × (n − r)!) With repetition: (n + r − 1)Cr
Unlike permutations, order does not matter in a combination, so the set A, B, C and the set C, B, A are counted as the same selection.
Worked Example
How many ways can you choose 6 numbers from 49 (a standard lottery)?
- nCr = 49! ÷ (6! × 43!)
- = (49 × 48 × 47 × 46 × 45 × 44) ÷ (6 × 5 × 4 × 3 × 2 × 1)
- = 13,983,816 combinations
So a single ticket has roughly a 1-in-14-million chance of matching all six numbers.
Combination Examples
| n | r | nCr | (n+r−1)Cr (with repeat) |
|---|---|---|---|
| 5 | 2 | 10 | 15 |
| 10 | 3 | 120 | 220 |
| 49 | 6 | 13,983,816 | 25,827,165 |
| 52 | 5 | 2,598,960 | 3,162,510 |
Where Combinations Are Used
Combinations count selections in lotteries and card hands, committee and team picks, menu and product bundles, and any probability problem where the arrangement is irrelevant. They are the heart of the binomial coefficient and Pascal’s triangle, and they pair with permutations to solve the full range of counting problems.
Frequently asked questions
What is a combination?+
A combination is a selection of items where order does not matter. Choosing 6 lottery numbers from 49 is a combination because the order you pick them in makes no difference to the ticket. The number of combinations is written nCr or 'n choose r'.
What is the formula for combinations?+
The number of combinations of r items chosen from n is nCr = n! ÷ (r! × (n − r)!). For 49 numbers chosen 6 at a time, that is 49! ÷ (6! × 43!) = 13,983,816 possible tickets.
What is the difference between a combination and a permutation?+
A combination counts selections where order is irrelevant, while a permutation counts arrangements where order matters. Choosing 3 fruits for a basket is a combination; arranging 3 books on a shelf is a permutation. For the same n and r, there are always r! times as many permutations as combinations.
What does combination with repetition mean?+
With repetition, the same item can be chosen more than once, such as picking 3 scoops of ice cream from 5 flavors where flavors can repeat. The count is (n + r − 1)Cr. This calculator shows both the standard and the with-repetition totals.
How many combinations are there in a lottery?+
It depends on the format. A 6-from-49 lottery has 49C6 = 13,983,816 combinations, so a single ticket has about a 1-in-14-million chance of matching all six numbers. Enter your lottery's n and r to get the exact count.