Calculator
Example Data Table
| n | r | Mode | Expression | Answer | Example use |
|---|---|---|---|---|---|
| 5 | 2 | Without repetition | C(5, 2) | 10 | Choose 2 people from 5 |
| 10 | 3 | Without repetition | C(10, 3) | 120 | Create 3 item groups |
| 4 | 3 | With repetition | C(6, 3) | 20 | Pick repeated flavors |
| 12 | 5 | Without repetition | C(12, 5) | 792 | Select team members |
Formula Used
For selections without repetition, the calculator uses:
C(n, r) = n! / (r! × (n - r)!).
It also uses symmetry:
C(n, r) = C(n, n - r).
This reduces long work when r is large.
For selections with repetition, it uses:
C(n + r - 1, r).
This counts choices where an item may be selected more than once.
The step method uses:
answer = answer × numerator ÷ denominator.
Each division is exact at that step.
How to Use This Calculator
- Enter the total number of available items in the n field.
- Enter the number of selected items in the r field.
- Choose whether repetition is allowed.
- Add optional item labels if you want sample selections.
- Set the number of step rows you want to display.
- Press Calculate to show the result below the header.
- Use CSV or PDF export to save the calculation.
Why Combination Steps Matter
Combinations count selections where order does not matter. They appear in probability, scheduling, sampling, testing, games, and planning. A step view is useful because it shows how every factor enters the answer. It also helps learners find mistakes before a final value is used. Many people know the final formula, yet they still need a clear path from n and r to the result.
How the Tool Supports Learning
This calculator accepts large values and reduces r when possible. That means C(n, r) is treated like C(n, n-r) when the second form is shorter. The method saves work and keeps the displayed steps cleaner. You can also choose combinations with repetition. That option is common when items may repeat, such as dessert choices or repeated color selections. The output shows the formula, substituted values, factorial view, compact product view, and final count.
Practical Uses
Teachers can create examples for lessons. Students can verify homework. Analysts can estimate sample groups. Project managers can count team selections. Product teams can count feature bundles. The same idea also supports lottery odds, seating groups, password pattern checks, and quality audits. The export buttons make records easy to keep. A CSV file fits spreadsheets. A PDF report is useful for printing, sharing, or attaching to notes.
Reading the Result
Start with the selected mode. Then compare the formula line with your inputs. Review the simplified r value when the calculator uses symmetry. Finally, read the step product. Each numerator term is paired with a denominator term. This prevents hidden work. It also makes the method easier to explain.
Good Input Habits
Use whole numbers only. Keep r between zero and n for standard combinations. For repetition mode, n must be at least one. Very large answers may contain many digits, so export them when you need exact records. When labels are entered, treat them as examples only. The count is based on n and r, not on the label text itself. Add a short note for each calculation. This makes later review easier. Save one report for final work and another for practice comparisons. Clear records reduce repeated checking.
FAQs
1. What is a combination?
A combination is a selection where order does not matter. Choosing A and B is the same as choosing B and A.
2. What does n mean?
n is the total number of available items. For example, if a class has 20 students, n is 20.
3. What does r mean?
r is the number of items selected from the total group. If you choose 3 students, r is 3.
4. When should I use repetition mode?
Use repetition mode when the same item can be selected more than once, such as repeated flavors, colors, or resources.
5. Why does the calculator reduce r?
It uses C(n, r) = C(n, n-r). This creates fewer multiplication steps and gives the same exact answer.
6. Can r be greater than n?
For standard combinations, no. For repetition mode, r can be greater than n because items may repeat.
7. What is exported in the CSV file?
The CSV includes inputs, formula details, the final answer, displayed steps, notes, and sample selections when available.
8. Are item labels required?
No. Labels are optional. They only help show sample selections. The main result depends on n, r, and mode.