Calculator Input
Example Data Table
These sample rows show common ordered-selection cases with and without repetition.
| Scenario | n | k | Mode | Formula | Result |
|---|---|---|---|---|---|
| Arrange 3 letters from 6 unique letters | 6 | 3 | Without repetition | 6! / 3! | 120 |
| Create a 4-digit code from 10 digits | 10 | 4 | With repetition | 10^4 | 10,000 |
| Choose podium positions from 8 runners | 8 | 3 | Without repetition | 8! / 5! | 336 |
| Assign 5 labeled slots from 12 items | 12 | 5 | Without repetition | 12! / 7! | 95,040 |
Formula Used
A k permutation counts ordered selections. The formula changes depending on whether items can repeat.
Without repetition:
P(n,k) = n! / (n-k)!
This works because you pick from n choices, then n−1, then n−2, until k positions are filled.
With repetition:
P(n,k) = n^k
This works because each of the k positions can use any of the n choices independently.
Use the non-repetition version for rankings, seat orders, and unique assignments. Use the repetition version for passwords, codes, and repeatable labeled selections.
How to Use This Calculator
- Enter the total number of available items as n.
- Enter the number of ordered positions as k.
- Select whether repetition is allowed.
- Choose how far the graph and table should extend.
- Optionally include the k = 0 row and worked steps.
- Press the calculate button to show the result above the form.
- Review the exact value, scientific notation, digits, chart, and table.
- Use the CSV or PDF buttons to save the output.
FAQs
1. What does k permutation mean?
A k permutation counts how many ordered arrangements can be formed by placing k items into positions. Order matters, so different sequences are counted separately.
2. What is the difference between permutation and combination?
Permutations care about order, while combinations ignore order. For example, ABC and BAC are different permutations but the same combination when only membership matters.
3. When should repetition be turned on?
Turn repetition on when the same item can appear more than once across positions. Common examples include lock codes, repeated symbols, and independent slot assignments.
4. Why does the chart use log10 values?
Permutation counts grow very quickly. A log10 scale keeps large results readable and lets you compare growth across k values without flattening smaller bars.
5. Can k be larger than n?
Yes, but only when repetition is allowed. Without repetition, you cannot fill more unique positions than the number of available unique items.
6. Why are exact values shown with scientific notation too?
Exact values are useful for precision, while scientific notation makes very large counts easier to scan, compare, and use in reports or quick reviews.
7. What happens when k equals zero?
The result is 1. There is exactly one way to arrange zero items: the empty arrangement. That is why the optional table can include a k = 0 row.
8. Where is this calculator useful in maths and practice?
It is useful in counting theory, probability, coding systems, ranking tasks, assignment problems, tournament ordering, password design, and sequence analysis.