Combination with Repetition (nHr)

Count how many ways to choose r items from n types when repetition is allowed and order does not matter.

Input

Count the ways to choose r items from n types when repetition is allowed (combination with repetition nHr).

Number of types n

How many distinct types to choose from (1 or more)

Number to choose r

How many to pick, repeats allowed (0 or more)

Ways to choose 3 from 5 types with repetition

nHr = 5H3 = (7)C3

Plain combination 5C3

The combination with repetition nHr equals the binomial of n plus r minus 1 choose r, matching the stars and bars arrangement.

A combination with repetition nHr counts the ways to choose r items from n types when repeats are allowed and order does not matter.
The formula is nHr equal to the binomial coefficient of n plus r minus 1 choose r, often called the stars and bars method.
Example: choosing 2 sweets from 3 kinds with repetition allowed gives 3H2 equal to 4 choose 2, which is 6 ways.
The plain combination nCr counts choices without repetition and is defined only when r is at most n.
Large results stay exact because the values are computed with integer big-number arithmetic.

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