Combinations with Repetition Calculator

This calculator counts combinations with repetition, the number of ways to choose a number of items from several types when you can pick the same type more than once and the order does not matter. Ordinary combinations assume you cannot repeat an item, but many real situations allow repeats: choosing a dozen doughnuts from six flavours, where you can take several of the same, picking scoops of ice cream from the available flavours, or distributing identical items among categories. Because repeats are allowed but order still does not matter, the count follows a different formula from ordinary combinations, derived by a neat argument known as stars and bars. This tool computes it. You enter the number of types to choose from and the number of items you are choosing, and the calculator returns the number of combinations with repetition, along with the count of arrangements where order does matter for comparison, and the values you entered. The results update as you type. Use it for probability and combinatorics homework, for counting selections where repeats are allowed, or to understand the difference from ordinary combinations. The formula is the combination of n plus r minus 1, choose r, where n is the number of types and r is the number you pick. This comes from the stars and bars method: imagine the r items as stars and use n minus 1 bars to divide them among the n types, then count the arrangements. The calculator also shows n to the power r, the count if order did matter, which is much larger, to highlight how requiring order multiplies the possibilities. This counting principle is a staple of combinatorics and appears throughout probability, where knowing how many ways something can happen is the first step.

Combinations with repetition = (n + r - 1) choose r, by the stars and bars method. Order does not matter and repeats are allowed. n to the power r counts ordered selections.