Calculator Inputs
Example Data Table
| Scenario | Item types n | Selections r | Formula | Result |
|---|---|---|---|---|
| Choose snacks | 5 | 3 | C(5 + 3 - 1, 3) | 35 |
| Choose ice cream scoops | 6 | 4 | C(6 + 4 - 1, 4) | 126 |
| Distribute tokens | 4 | 7 | C(4 + 7 - 1, 7) | 120 |
| Select lab samples | 8 | 2 | C(8 + 2 - 1, 2) | 36 |
Formula Used
The calculator uses the standard combination with repetition formula:
C(n + r - 1, r) = (n + r - 1)! / r!(n - 1)!
Here, n is the number of item types. r is the number of selections.
The formula is also equal to C(n + r - 1, n - 1).
This is called the stars and bars method. Stars represent selected items. Bars separate item categories. The calculator also compares the answer with ordinary combinations and ordered selections.
How to Use This Calculator
- Enter the number of item types in the n field.
- Enter the number of selections in the r field.
- Add item labels if you want sample combinations.
- Set a sample display limit for the preview table.
- Click Calculate to show the result above the form.
- Use CSV or PDF buttons to save the output.
Understanding Repeated Combinations
A combination with repetition counts selections where order does not matter, and items may repeat. It is also called a multiset count. Think about choosing scoops from flavors. Vanilla, vanilla, chocolate is one valid selection. Chocolate, vanilla, vanilla is not a new one.
This calculator uses the classic stars and bars method. The method changes repeated choices into separator positions. If there are n item types and r selections, the count is C(n + r - 1, r). The same value also equals C(n + r - 1, n - 1).
Why This Count Matters
Repeated combinations appear in many practical tasks. They help plan product bundles, menu choices, token distributions, lab mixtures, and password style patterns. They also support probability lessons and discrete math examples. Because order is ignored, the answer is often far smaller than ordered sequences.
For example, choosing three snacks from five snack types gives C(7, 3). The answer is thirty five. This includes triples like chips, chips, juice. It also includes chips, juice, nuts. It does not count juice, chips, nuts again, because the same group already exists.
Advanced Use
Use exact values for classroom work, reports, and planning. Compare the repeated result with a no repetition result. This shows how allowing repeated items changes the space. Also compare it with ordered selections with replacement. That value is n raised to r, and it grows much faster.
The optional labels field helps create sample combinations. Keep labels short. Use commas between names. Listing is best for small totals only. Large counts can produce thousands of rows, so the tool limits the preview.
Reading the Results
The result card gives the main count, formula substitution, digits, and related values. The stars and bars value shows the total number of positions used before choosing separators. The probability line gives the chance of one specific combination under a uniform combination model.
Always check the meaning of your problem first. Use this calculator when order is not important. Use a permutation tool when order matters. Use ordinary combinations when repeats are not allowed. Clear assumptions create correct counting.
Save export files for later review and sharing during study.
FAQs
1. What is a combination with repetition?
It is a count of selections where order does not matter, and the same item type can be chosen more than once.
2. What formula does this calculator use?
It uses C(n + r - 1, r), where n is item types and r is selections. This is the stars and bars formula.
3. When should I use this calculator?
Use it when you select groups from categories, repeats are allowed, and order does not create a new result.
4. What does n mean?
The value n means the number of different item types, categories, labels, flavors, products, or options available.
5. What does r mean?
The value r means the total number of items selected from the available item types, while allowing repeated choices.
6. Is order counted in the result?
No. The result ignores order. A, A, B is the same combination as A, B, A in this counting method.
7. Why is the sample list limited?
Repeated combinations can grow very quickly. The limit keeps the page fast, readable, and useful for small demonstrations.
8. Can I download my result?
Yes. Use the CSV button for spreadsheet work. Use the PDF button for a simple printable result summary.