# Combinations With Repetition Calculator

URL: https://speedycalculator.com/calculators/statistics/combinations-with-repetition/
Last updated: 2026-10-30

> This calculator counts combinations with repetition (multisets) – the number of ways to choose r items from n types when the same type may be taken more than once and order does not matter, C(n + r − 1, r) – and lists them for small cases.

## Inputs

- Number of types to choose from (n) — Ignored when you type the objects themselves below.
- Number of items chosen (r), repeats allowed
- Objects to list (optional, comma-separated) — If you type names here, n is taken from how many you entered and the list below uses them.
- List the selections

## Results

- Combinations with repetition
- Chance of one particular selection (if all equally likely)
- Without repetition, for comparison
- Ordered sequences with repetition (nʳ)
- Note
- All selections

## Formula

```
C′(n, r) = C(n + r − 1, r) = (n + r − 1)! ÷ (r! × (n − 1)!)
```

## Explanation

Sometimes you can pick the same kind of item more than once: three scoops of ice cream from five flavours, a dozen doughnuts from six varieties or five coins from four denominations. Order still does not matter – chocolate-vanilla-chocolate is the same cup as chocolate-chocolate-vanilla. These are combinations with repetition, also called multisets.

The count comes from the "stars and bars" argument: picture the r chosen items as stars and separate the n types with n − 1 bars. Every arrangement of r stars and n − 1 bars is one selection, so the count is C(n + r − 1, r). Three scoops from five flavours gives C(7, 3) = 35, compared with only C(5, 3) = 10 if all scoops had to differ.

Type the item types (flavours, colours, coins) or just their number, choose r and the calculator returns the exact count, the comparison with ordinary combinations and, for up to 300 results, the full list of selections with repeats shown.

### How to use the Combinations With Repetition Calculator

- **Number of types to choose from (n)** — Ignored when you type the objects themselves below.
- **Number of items chosen (r), repeats allowed**
- **Objects to list (optional, comma-separated)** — If you type names here, n is taken from how many you entered and the list below uses them.
- **List the selections**

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

n + r − 1 positions are filled with r stars and n − 1 bars; the number of ways to choose where the stars go is the binomial coefficient C(n + r − 1, r). Listing generates non-decreasing sequences of types.

### Worked example

Five flavours, three scoops, repeats allowed: C(5 + 3 − 1, 3) = C(7, 3) = 7 × 6 × 5 ÷ 6 = 35 cups. They split into 5 cups of a single flavour, 20 with one flavour doubled (5 × 4) and 10 with three different flavours. 5 + 20 + 10 = 35.

### With vs without repetition

nrWithout repetition C(n, r)With repetition C(n + r − 1, r)531035612—6,1881052522,00226565,780142,506

## Worked example

Inputs — Number of types to choose from (n): 5; Number of items chosen (r), repeats allowed: 3; Objects to list (optional, comma-separated): Vanilla, Chocolate, Strawberry, Mint, Mango; List the selections: Yes – show up to 300.

Results — **Combinations with repetition = 35**; Chance of one particular selection (if all equally likely) = 1 in 35 (2.857143%); Without repetition, for comparison = 10; Ordered sequences with repetition (nʳ) = 125; Note = n = 5 (counted from the objects you listed); All selections = {Vanilla, Vanilla, Vanilla} {Vanilla, Vanilla, Chocolate} {Vanilla, Vanilla, Strawberry} {Vanilla, Vanilla, Mint}
{Vanilla, Vanilla, Mango} {Vanilla, Chocolate, Chocolate} {Vanilla, Chocolate, Strawberry} {Vanilla, Chocolate, Mint}
{Vanilla, Chocolate, Mango} {Vanilla, Strawberry, Strawberry} {Vanilla, Strawberry, Mint} {Vanilla, Strawberry, Mango}
{Vanilla, Mint, Mint} {Vanilla, Mint, Mango} {Vanilla, Mango, Mango} {Chocolate, Chocolate, Chocolate}
{Chocolate, Chocolate, Strawberry} {Chocolate, Chocolate, Mint} {Chocolate, Chocolate, Mango} {Chocolate, Strawberry, Strawberry}
{Chocolate, Strawberry, Mint} {Chocolate, Strawberry, Mango} {Chocolate, Mint, Mint} {Chocolate, Mint, Mango}
{Chocolate, Mango, Mango} {Strawberry, Strawberry, Strawberry} {Strawberry, Strawberry, Mint} {Strawberry, Strawberry, Mango}
{Strawberry, Mint, Mint} {Strawberry, Mint, Mango} {Strawberry, Mango, Mango} {Mint, Mint, Mint}
{Mint, Mint, Mango} {Mint, Mango, Mango} {Mango, Mango, Mango}
All 35 listed..

## Key terms

- **Multiset**: A collection where elements may repeat but order does not matter.
- **Stars and bars**: A counting method that turns multisets into arrangements of stars and dividers.
- **Type**: One of the n kinds of item that can be chosen.

## FAQ

**How many ways can I choose 3 scoops from 5 flavours?**
With repeats allowed, C(7, 3) = 35; with all different flavours, C(5, 3) = 10.

**What is the formula for combinations with repetition?**
C(n + r − 1, r) = (n + r − 1)! ÷ (r!(n − 1)!).

**Why is it called stars and bars?**
The r items are drawn as stars and n − 1 bars split them into the n types; each arrangement is a selection.

**Can r be larger than n?**
Yes. A dozen doughnuts from 6 varieties is C(17, 12) = 6,188.

**How many ways can I pick 5 digits if repeats are allowed and order does not matter?**
C(14, 5) = 2,002.

**Are all selections equally likely?**
Not if you draw items one by one at random – {A, A} is less likely than {A, B} – so the "chance" output assumes each selection is equally likely.


## Sources

- [NIST Digital Library of Mathematical Functions – Combinatorial analysis](https://dlmf.nist.gov/26)
- [Wolfram MathWorld – Probability and statistics](https://mathworld.wolfram.com/topics/ProbabilityandStatistics.html)