Calculator
Formula Used
Without repetition: V = n! / (n - r)! . Each selected item is removed from the next position.
With repetition: V = n^r . Each position can use any item again.
Probability option: Probability = favorable ordered outcomes / total variations.
How to Use This Calculator
- Enter the number of available distinct items as n.
- Enter the number of ordered positions as r.
- Choose whether items can repeat in the sequence.
- Add favorable ordered outcomes when you need a probability estimate.
- Press the calculate button and read the result below the header.
- Use CSV or PDF download for reports and records.
Example Data Table
| Use Case | n | r | Type | Formula | Result |
|---|---|---|---|---|---|
| Three winners from ten people | 10 | 3 | Without repetition | 10! / 7! | 720 |
| Four digit passcode from ten digits | 10 | 4 | With repetition | 10^4 | 10,000 |
| Five seat order from eight guests | 8 | 5 | Without repetition | 8! / 3! | 6,720 |
About Calculating Variations
Calculating variations helps you count ordered choices. Order matters in each result. A code, seat plan, password, route, or task list can use variation logic. The calculator handles two common cases. One case allows repeated items. The other case uses every item once at most.
Why Order Matters
A variation is different from a combination. In combinations, AB and BA are the same group. In variations, AB and BA are different outcomes. This difference is important when ranks, positions, steps, or slots have meaning. It also matters when choices create a sequence.
Using The Inputs
Start with the number of available items. This value is usually called n. Then enter the number of selected positions. This value is usually called r. Choose whether items can repeat. The result area shows the exact count, digit count, scientific form, and a comparison value. You can also enter favorable ordered outcomes. The tool then estimates the chance of those outcomes.
Practical Examples
Variation counting is useful in many general tasks. A teacher can count possible seating orders. A coach can test lineup choices. A designer can count menu paths. A planner can compare coded labels. A student can verify homework steps. The same idea also supports lottery studies, schedule checks, and simple probability models.
Interpreting Results
Large variation counts grow fast. Repetition makes them grow even faster. Scientific notation keeps the display readable. The log value helps compare very large answers. Exact digits are useful for reports when the result is not too long. The formula section explains the logic so you can audit each answer.
Good Habits
Check that r does not exceed n when repetition is not allowed. Use repetition only when the same item can appear in several positions. Review the example table before entering your own values. Export the report when you need records for class, planning, or documentation.
Checking The Method
Use the chosen rule before trusting a count. For no repeats, each used item reduces the next slot. For repeats, every slot keeps all choices open. This is why two similar inputs can create very different totals. Save both values when comparing plans or explaining a decision. Keep assumptions clear in every exported report.
FAQs
What is a variation?
A variation is an ordered selection. The same items in a different order create a different result.
When should I use no repetition?
Use no repetition when an item cannot appear twice. Examples include seating guests, ranking winners, or assigning unique tasks.
When should I use repetition?
Use repetition when the same item can be used again. Passcodes, product codes, and repeated route choices often allow repetition.
Why must r be smaller than n without repetition?
Without repetition, every chosen position consumes one item. You cannot fill more unique positions than available items.
Does order always matter here?
Yes. This calculator is for ordered arrangements. If order does not matter, use a combination calculator instead.
What does the scientific form mean?
Scientific form gives a compact display for large counts. It shows a decimal value multiplied by a power of ten.
Can I calculate probability?
Yes. Enter favorable ordered outcomes. The calculator divides them by total variations and returns an estimated percentage.
Why is the exact number sometimes long?
Variation counts grow quickly. More items and positions create very large totals, especially when repetition is allowed.