Find repeated-selection counts with exact values and charts. Review factorial steps, logs, and growth patterns. Compare item types and picks clearly for better decisions.
For combinations with repetition, the count of ways to choose r items from n distinct types is:
C(n + r - 1, r)
Equivalent form: C(n + r - 1, n - 1)
Factorial form: (n + r - 1)! / [r! × (n - 1)!]
This comes from the stars and bars method. Think of r chosen items as stars and n - 1 separators as bars.
Every arrangement of those stars and bars maps to one multiset, so counting the arrangements gives the answer.
| Item Types (n) | Picks (r) | Formula | Result |
|---|---|---|---|
| 3 | 4 | C(6,4) | 15 |
| 4 | 3 | C(6,3) | 20 |
| 5 | 2 | C(6,2) | 15 |
| 6 | 4 | C(9,4) | 126 |
The last row becomes dynamic after calculation and mirrors your current input values.
It counts selections where you can reuse the same type more than once, but arrangement order is ignored. It is the standard model for multisets.
Use it when repeated choices are allowed. Ordinary combinations assume each item can appear at most once, while this model allows duplicates.
Stars represent chosen items and bars separate categories. Counting their arrangements converts a repeated-choice problem into a standard combination problem.
You place r stars and n - 1 bars into one line. That creates n + r - 1 total positions, then you choose which positions hold the stars or bars.
Yes. Even moderate values can create huge integers. That is why this page shows exact output, scientific notation, digit count, and a log-based growth graph.
Then this is not the right formula. Order-sensitive repeated selections usually require a different counting model, such as sequences or permutations with repetition.
It helps with inventory bundles, candy selection, symbolic algebra terms, occupancy distributions, ballot models, and many discrete mathematics exercises.
Repeated-choice counts grow quickly. A log10 scale keeps the curve readable and lets you compare small and very large outputs on one plot.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.