Combinations with Repetition Calculator

When you can pick the same item more than once, use combinations with repetition.

How the Math Works

The Combinations with Repetition Calculator uses the formula C(n + r - 1, r) to count the number of ways to select r items from n types where repetition is allowed and order doesn't matter. This formula is derived from the 'stars and bars' method in combinatorics: imagine r identical items (stars) that need to be divided into n distinct groups (bins) using n-1 dividers (bars). The total number of positions is n + r - 1, and we choose r positions for the stars (or equivalently n-1 positions for the bars). For example, selecting 3 scoops of ice cream from 5 flavors requires C(5 + 3 - 1, 3) = C(7, 3) = 35 combinations.

Practical Applications

To apply this calculation, first identify your two parameters: n represents the number of distinct item types or categories available, and r represents the number of items you're selecting total. The formula works regardless of whether you're choosing with or without replacement since repetition is explicitly allowed. For instance, if you need to distribute 10 identical candies among 4 children, you'd calculate C(4 + 10 - 1, 10) = C(13, 10) = 286 ways. This approach also applies to polynomial expansions, where finding the number of terms in (a + b + c)^r uses the same formula.

Day-to-Day Use

This calculation helps in everyday situations involving distribution and selection with unlimited supply. When planning a menu with unlimited portions from 6 dish options for 4 guests, you can determine there are C(6 + 4 - 1, 4) = 126 different meal combinations. It's useful for inventory planning—like figuring out how many ways to package 15 identical items into 3 different box sizes—and for understanding probability in games where you draw multiple items from the same set repeatedly, such as drawing scoops from an ice cream tub or selecting sprinkles for a dessert.

Worked example

Choosing 3 from 5 with repetition gives 35.

FAQ