Two dice
- Input
- CR(6, 2)
- Expected output
- 21
Each pair of dice values counts once, including the 6 repeated pairs like (3,3); without repetition there would only be 15 distinct pairs.
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combinations with repetition CR(n,r)
CR(n,r) = C(n+r-1, r) counts groupings where the same element can repeat and order does not matter, unlike a classic lottery draw, where each number comes out at most once. The tool adds n+r-1 before applying the same simple-combination formula and returns the exact BigInt result, with no rounding.
Each pair of dice values counts once, including the 6 repeated pairs like (3,3); without repetition there would only be 15 distinct pairs.
2 dividers added to the 3 scoops fill 7 possible slots among the 5 flavors; the same choice without repeating a flavor would only have 10 combinations.
Only possible because repetition allows picking more items than categories exist; the no-repetition version, C(4,10), returns an error in this same calculator.
A combination is a choice of r elements out of n where order does not matter. The formula is C(n, r) = n! / (r! · (n − r)!), read "n choose r". Example: picking 2 fruits from apple, pear and grape gives C(3, 2) = 3 groups (apple and pear, apple and grape, pear and grape). In the calculator, enter n and r and the Combination mode shows the exact result and the working step by step.
Yes, and that is exactly where it differs from a simple combination: CR(4,10) = 286 is a valid result because repetition removes the r ≤ n limit, while C(4,10) without repetition returns an error in this same tool for not having enough items to choose without repeating.
C(n,r) picks r distinct items from n categories, never repeating one; CR(n,r) allows repeating any category as many times as fits in r. For the same 6 die values picked 2 at a time, that is the difference between 15 combinations (C) and 21 combinations (CR).
Yes, for the same n and r with r ≤ n: allowing repetition only adds possibilities, it never removes any of the ones that already existed without repeating. That shows up in CR(6,2) = 21 versus C(6,2) = 15, a difference of exactly 6 repeated pairs.
Choose r of n, no order. E.g. team, lottery.
C(n, r) = n! / (r! · (n − r)!)
120
Multiplies the total. Use 1 (Powerball: 26).
Click a cell to use n and k.
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