Combinations With Repetition
Calculate the number of combinations with repetition (multichoose) for any n types and r choices. Free online combinatorics calculator with formula breakdown and growth chart.
About This Calculator
Combinations with repetition, also known as multichoose or stars-and-bars, count the number of ways to select r items from n distinct types when items can be chosen more than once and order does not matter. This is a fundamental concept in combinatorics with applications ranging from counting ice cream flavor combinations to solving Diophantine equations and analyzing voting patterns.
The formula for combinations with repetition is C'(n,r) = C(n+r-1,r) = (n+r-1)!/(r!(n-1)!). This is equivalent to the binomial coefficient of (n+r-1) choose r. The derivation comes from the stars-and-bars method: arrange r identical stars (representing chosen items) and n-1 bars (representing dividers between types), and count the number of distinct arrangements. Each arrangement corresponds to a unique combination with repetition.
For example, choosing 3 scoops of ice cream from 5 available flavors (chocolate, vanilla, strawberry, mango, pistachio) where you can pick the same flavor multiple times gives C'(5,3) = C(7,3) = 35 different combinations. Without repetition, you would only have C(5,3) = 10 combinations. This calculator supports both small and large values — simply enter n (the number of types) and r (the number of choices), and it computes the result instantly with a visual growth chart showing how the count scales with r.
Frequently Asked Questions
What is combinations with repetition?
Combinations with repetition (also called multichoose or stars-and-bars) count the number of ways to select r items from n types where order does not matter and items can be chosen more than once. The formula is C'(n,r) = C(n+r-1,r) = (n+r-1)!/(r!(n-1)!).
How is combinations with repetition different from combinations without repetition?
In combinations without repetition, each item can be chosen only once (nCr). In combinations with repetition, items can be chosen multiple times. For example, choosing 3 scoops of ice cream from 5 flavors allows repeated flavors, giving C(5+3-1,3)=35 combinations, while without repetition only C(5,3)=10.
What is the formula for combinations with repetition?
The formula is C'(n,r) = C(n+r-1,r) = (n+r-1)!/(r!(n-1)!). This is equivalent to the binomial coefficient of (n+r-1) choose r. The formula is derived from the stars-and-bars method in combinatorics.
How many combinations with 5 numbers with repetition are possible?
There are 2,002 possible combinations with repetition of 5 numbers from digits 0-9 (n=10, r=5). This uses the formula C'(10,5) = C(14,5) = 2,002. In contrast, there are only 252 combinations without repetition and 100,000 permutations with repetition.
What are real-world applications of combinations with repetition?
Real-world applications include counting ice cream or pizza topping combinations where flavors can repeat, distributing identical items into distinct boxes (stars-and-bars problems), counting multisets, analyzing voting patterns, solving Diophantine equations, and modeling certain probability problems.
What is the stars-and-bars method?
The stars-and-bars method visualizes combinations with repetition by arranging r stars (items) and n-1 bars (dividers between types). The total number of arrangements is C(n+r-1,r). For example, choosing 3 scoops from 5 flavors is like arranging 3 stars and 4 bars.
Is this calculator free to use?
Yes, all calculators on Calculy are completely free to use with no hidden fees, subscriptions, or usage limits. You can perform unlimited calculations.
What is the difference between combinations with repetition and permutations with repetition?
In combinations with repetition, order does not matter — choosing chocolate, vanilla, chocolate is the same as vanilla, chocolate, chocolate. In permutations with repetition, order matters — each sequence is counted separately. The formula for permutations with repetition is n^r, which gives much larger numbers. For example, with n=5 and r=3, combinations = 35 but permutations = 125.