Combination with Replacement Calculator

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Combinations Replacement C^R(n,r)

\[ C^R(n,r) = \frac{(n + r - 1)!}{ r! (n - 1)! } \]

n (objects) =

r (sample) =

Answer:

= 35

Solution:

\[ C^R(n,r) = \frac{(n + r - 1)!}{ r! (n - 1)! } \]\[ C^R(n,r) = C^R(5,3) \]\[ = \frac{(5 + 3 - 1)!} {3! ( 5 - 1)!} \]

= 35

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Calculator Use

For a combination replacement sample of r elements taken from a set of n distinct objects, order does not matter and replacements are allowed.

The Combinations Replacement Calculator will find the number of possible combinations that can be obtained by taking a subset of items from a larger set. Replacement or duplicates are allowed meaning each time you choose an element for the subset you are choosing from the full larger set. This calculates how many different possible subsets can be made from the larger set. For this calculator, the order of the items chosen in the subset does not matter.

Combinations with replacement, also called multichoose, for C^R(n,r) = C(n+r-1,r) = (n+r-1)! / r! (n+r-1 - r)! = (n+r-1)! / r! (n - 1)!. For n >= 0, and r >= 0. If n = r = 0, then C^R(n,r) = 1.

Factorial: There are n! ways of arranging n distinct objects into an ordered sequence, permutations where n = r.
Combination: The number of ways to choose a sample of r elements from a set of n distinct objects where order does not matter and replacements are not allowed.
Combination Replacement: The number of ways to choose a sample of r elements from a set of n distinct objects where order does not matter and replacements are allowed.
n: the set or population
r: subset of n or sample set

Combination with Replacement Formula:

\( C^R(n,r) = \dfrac{(n + r - 1)!}{ r! (n - 1)! } \)

References

For more information on combinations and binomial coefficients please see Wolfram MathWorld: Combination.