Calculator
Choose a counting mode, enter your values, and compute valid outcomes with optional invalid-case subtraction.
Example Data Table
These sample rows show common counting situations and the matching formulas.
| Scenario | Inputs | Formula | Result | Notes |
|---|---|---|---|---|
| Outfit planning | 3 shirts, 4 pants, 2 shoes | 3 × 4 × 2 | 24 | Independent stage choices. |
| Seat awards | n = 8, r = 3 | 8P3 | 336 | Order matters. |
| Committee picks | n = 8, r = 3 | 8C3 | 56 | Order does not matter. |
| PIN design | n = 10, r = 4 | 10^4 | 10,000 | Repetition allowed. |
| Bracelet layout | n = 6 | (6 - 1)! / 2 | 60 | Mirror images identical. |
| Letter arrangements | 2, 2, 1 | 5! / (2! × 2! × 1!) | 30 | Repeated letters included. |
Formula Used
Multiplication principle: Multiply the choices for each independent stage. Formula: a × b × c × ....
Permutations: Count ordered selections without repetition. Formula: nPr = n! / (n-r)!.
Combinations: Count unordered selections without repetition. Formula: nCr = n! / (r!(n-r)!).
Repeated sequences: Count ordered selections with repetition allowed. Formula: n^r.
Circular arrangements: Count distinct rotations. Formula: (n-1)!. If reflections are identical, use (n-1)! / 2.
Multiset permutations: Count arrangements with repeated items. Formula: n! / (a!b!c!...).
How to Use This Calculator
- Select the counting mode that matches your problem structure.
- Enter stage counts, n and r values, or repeated group sizes.
- Optionally enter invalid outcomes already counted in the raw total.
- For circular cases, choose whether mirror images should match.
- Press Calculate Now to show the result above the form.
- Review the chart, summary table, and formula explanation.
- Download CSV or PDF files for documentation or sharing.
Frequently Asked Questions
1. What does the fundamental counting principle mean?
It says you multiply the number of choices from independent stages. If one step has 3 options and another has 4, there are 12 total outcomes.
2. When should I use permutations?
Use permutations when order matters. Ranking winners, assigning seats, and creating nonrepeating access codes are common examples.
3. When should I use combinations?
Use combinations when order does not matter. Selecting committee members or choosing toppings without caring about arrangement fits this rule.
4. Why is there an invalid outcomes field?
Some problems start with a broad count, then remove disallowed cases. This field lets you subtract those cases without rebuilding the whole expression.
5. What is a repeated sequence?
A repeated sequence allows the same choice to appear again in another position. Four-digit PIN codes with repeated digits are a standard example.
6. Why do circular arrangements use (n-1)!?
Rotating a circular arrangement does not create a new layout. Fixing one object removes equivalent rotations, leaving (n-1)! distinct arrangements.
7. What are multiset permutations?
They count arrangements when identical items repeat. For example, the letters in AABBC do not all create unique rearrangements because some letters match.
8. Can this calculator handle large values?
Yes, but very large counts may display in scientific notation. That keeps results readable when factorials or repeated powers grow extremely quickly.