Calculator Input
Plotly Graph
This graph compares logarithmic growth for combination and permutation values across changing selection sizes.
Example Data Table
| Scenario | n | r | Repetition | Combination Result | Permutation Result |
|---|---|---|---|---|---|
| Select 3 students from 10 | 10 | 3 | No | 120 | 720 |
| Create 4-digit codes from 5 symbols | 5 | 4 | Yes | 70 | 625 |
| Choose 2 prizes from 8 | 8 | 2 | No | 28 | 56 |
| Pick 5 scoops from 6 flavors | 6 | 5 | Yes | 252 | 7,776 |
Formula Used
Combination Without Repetition
C(n, r) = n! / [r!(n-r)!]. Order does not matter here.
Permutation Without Repetition
P(n, r) = n! / (n-r)!. Order matters here.
Combination With Repetition
C(n+r-1, r) = (n+r-1)! / [r!(n-1)!]. Repeats are allowed.
Permutation With Repetition
n^r. Each position may reuse any available item.
How to Use This Calculator
- Enter the total available items as n.
- Enter the number of chosen positions as r.
- Select whether repetition is allowed.
- Choose the preferred calculation mode.
- Click the calculate button.
- Review exact values, digit counts, and scientific notation.
- Use export buttons for CSV or PDF copies.
- Inspect the graph to compare value growth patterns.
Frequently Asked Questions
1. What is the main difference between combinations and permutations?
Combinations ignore order. Permutations count different orders separately. Choosing three winners is a combination problem. Assigning gold, silver, and bronze is a permutation problem because positions matter.
2. When should I allow repetition?
Allow repetition when the same item can appear again. Password symbols, repeated survey choices, and ice cream flavor scoops often fit this case. Team selection usually does not.
3. Why do values grow so quickly?
Counting formulas use multiplication across many terms. As n or r rises, the number of possible arrangements expands very fast. That is why large outcomes often need scientific notation.
4. Can this tool handle large values?
Yes. This page uses string-based arithmetic for exact integer results. It avoids normal integer overflow for many practical counting cases and also shows digit counts for large outputs.
5. Why is r limited by n without repetition?
Without repetition, each selected item must be unique. You cannot choose more unique items than the total number available. That makes r greater than n invalid in that setting.
6. What does scientific notation help me understand?
Scientific notation makes huge answers easier to read and compare. It shows the leading digits and the power of ten, which helps you judge scale quickly.
7. Why does the graph use logarithmic values?
Permutation and combination outputs can become enormous. Logarithmic plotting compresses large ranges into a readable chart. This makes growth trends much easier to compare visually.
8. What real applications use these formulas?
They appear in probability, survey design, coding theory, lottery analysis, scheduling, password counting, roster planning, and sampling problems. These formulas support many practical statistical decisions.