Permutation and Combination Calculator
Calculate factorials, permutations, combinations, and repetition-based counting results from the total items and selected items. Permutation and Combination Calculator helps students, teachers, researchers, and analysts solve counting problems quickly and accurately.
Highlights
- Calculate permutations and combinations quickly and accurately.
- Find factorials for total and selected items.
- Compare ordered and unordered selections without repetition.
- Calculate permutations and combinations with repetition allowed.
- Ideal for students, teachers, researchers, and analysts.
Table of Contents
How Permutation and Combination Calculator Works?
The calculator uses factorials to count arrangements and selections from n distinct total items, where r items are selected. A factorial multiplies every positive integer up to a given number.
The calculator reports n!, r!, and (n − r)! as intermediate values. A permutation counts ordered arrangements without repetition, so changing the order creates a different result.
A combination counts selections without considering order. Selecting A, B, and C is therefore the same combination as selecting C, B, and A.
When repetition is allowed, the formulas change. Ordered selections with repetition use n choices for each of r positions, while unordered selections with repetition use the multiset combination formula.
For example, with n = 10 and r = 3, the calculator produces the following results:
| Output | Calculation | Result |
|---|---|---|
| Factorial of n | 10! | 3,628,800 |
| Factorial of r | 3! | 6 |
| Factorial of n − r | 7! | 5,040 |
| Permutations | 10P3 | 720 |
| Combinations | 10C3 | 120 |
| Permutations with repetition | 103 | 1,000 |
| Combinations with repetition | 12C3 | 220 |
The inputs are whole numbers. Total items n can range from 1 to 1,000, while selected items r can range from 0 to 1,000. For permutations and combinations without repetition, the standard interpretation requires r ≤ n.
Practical Use Cases
Students can compare permutations and combinations while solving probability, algebra, and discrete mathematics exercises. For example, the calculator can determine how many ordered podium outcomes are possible from a group of competitors and how many three-person teams can be selected from the same group.
Teachers can use the calculator to prepare worked examples and demonstrate the difference between order-sensitive and order-insensitive counting. The intermediate factorial outputs also make it easier to show how each formula is constructed.
Researchers can calculate the number of possible experimental arrangements, sample selections, or repeated-choice patterns when designing studies involving distinct options and fixed selection counts.
Analysts can evaluate possible schedules, rankings, codes, or item selections. The repetition-based outputs are useful when an option may be selected more than once, while the standard permutation and combination results apply when each item can be used only once.
Accuracy and Limitations
The calculator applies exact factorial, permutation, and combination formulas to integer inputs within the supported ranges: n from 1 to 1,000 and r from 0 to 1,000. Permutations and combinations without repetition are mathematically valid only when r ≤ n; entering a larger r does not represent a valid selection from the available items. Factorials and counting results grow extremely rapidly, so very large inputs may produce values that are difficult to display or may exceed the numeric precision supported by the calculator implementation.
Quick Examples
Example #1: Small Team Arrangements
Calculate arrangements and selections when choosing 3 people from a group of 10.
Example #2: Committee Selection
Determine the number of ways to select and arrange 5 members from 20 candidates.
Example #3: Password Character Choices
Calculate counting results for selecting 4 positions from 8 available distinct characters.
