Permutation and Combination Calculator
Calculate permutations and combinations with our advanced calculator. Learn the difference between permutations and combinations with detailed explanations and examples.
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Examples
Permutation Example
If you have 5 different books and want to arrange 3 of them on a shelf, the number of possible arrangements is P(5,3) = 5! / (5-3)! = 5! / 2! = 60.
Combination Example
If you have 10 people and want to select a committee of 3, the number of possible committees is C(10,3) = 10! / (3! × (10-3)!) = 10! / (3! × 7!) = 120.
Permutation with Repetition
If you have 3 flavors of ice cream and want to create a 2-scoop cone where you can repeat flavors, the number of possible cones is 3² = 9.
Frequently Asked Questions
Permutation and combination are both ways of selecting items from a larger set, but they differ in whether order matters. In permutation, the order of selection matters (e.g., AB is different from BA). In combination, the order doesn’t matter (e.g., AB is the same as BA). For example, when selecting people for a committee, we use combinations because the order doesn’t matter. When arranging people in a line, we use permutations because the order matters.
Use permutation when the order or arrangement of items matters. Examples include: arranging books on a shelf, selecting a president, vice-president, and secretary from a group, or creating passwords where the order of characters matters. Use combination when only the selection matters, not the order. Examples include: selecting a committee from a larger group, choosing lottery numbers (in most games), or picking fruits from a basket.
“With repetition” means that items can be selected more than once. For example, when creating a 3-digit PIN code, you can use the same digit multiple times (e.g., 111). Without repetition means each item can only be selected once (e.g., when selecting 3 different people from a group). The formulas change significantly: with repetition, permutation is n^r and combination is C(n+r-1, r), while without repetition, permutation is n!/(n-r)! and combination is n!/(r! × (n-r)!).
Factorials grow very quickly, and calculating them manually for large numbers is impractical. For example, 20! is approximately 2.43 × 10^18. Our calculator handles these calculations automatically. For very large numbers, you can use logarithms or Stirling’s approximation: n! ≈ √(2Ï€n) × (n/e)^n. In practice, most calculators and computer programs use specialized algorithms to handle large factorials without causing overflow errors.
Standard permutations and combinations are defined only for non-negative integers, where n ≥ r ≥ 0. However, the gamma function extends the factorial function to complex numbers (except negative integers), allowing for generalized calculations. In practice, most real-world applications only involve non-negative integers. Our calculator is designed for these standard cases and will show an error if you try to calculate with invalid inputs.