Combination Selection
Permutations: 720, Combinations: 120
Combination Calculator
Calculate combinations C(n,r).
Loading calculator…
Results
Permutations P(n,r)
Combinations C(n,r)
What is this calculator for?
The Combination Calculator performs accurate combination calculations for academic, professional, and personal mathematics needs with instant results.
Formula
P(n,r) = n! / (n-r)!
C(n,r) = n! / (r! * (n-r)!)
Worked Examples
About This Calculator
The Combination Calculator computes the number of ways to select and arrange items from a set, supporting combination scenarios. Permutations count arrangements where order matters, while combinations count groups where order does not matter. These calculations are essential in probability theory, statistical analysis, lottery odds, quality control, and many other fields where counting possibilities is required.
Enter the total number of items in your set and the number of items you want to select. The Combination Calculator instantly calculates both permutations and combinations, giving you the complete picture for combination probability and counting problems. The results update immediately as you adjust either value.
The Combination Calculator handles both permutations and combinations simultaneously, giving you a complete picture of your counting problem. Understanding the difference between ordered and unordered selections is crucial in combination probability and statistics work. This calculator helps you avoid the common mistake of using the wrong formula for your specific scenario.
Frequently Asked Questions
When should I use permutations versus combinations for combination?
Use permutations when the order of selection matters, such as arranging items, forming passwords, or ranking candidates. Use combinations when order does not matter, such as selecting a committee, choosing lottery numbers, or picking items from a menu. The key question is whether rearranging the same items creates a different combination outcome.
Why are permutations always greater than or equal to combinations for combination?
Permutations count every possible arrangement of the selected items, while combinations count each unique group only once. Since each combination of r items can be arranged in r factorial different ways, permutations equal combinations multiplied by r factorial. This relationship is fundamental to combination counting problems.
What is the difference between permutations with and without repetition for combination?
Without repetition, each item can be selected only once. With repetition, items can be reused. For example, a 4-digit PIN allows repetition, while selecting 4 cards from a deck does not. The Combination Calculator computes permutations and combinations without repetition for accurate combination results.
How are permutations and combinations used in combination probability?
To find the probability of a specific outcome, divide the number of favorable arrangements by the total number of possible arrangements. This approach solves problems ranging from card games to quality control, making combination probability calculations systematic and accurate.
Can I use the Combination Calculator for lottery odds calculations?
Yes, lottery odds are calculated using combinations. For example, choosing 6 numbers from 49 gives over 13.9 million possible combinations, meaning the odds of matching all six are approximately 1 in 14 million. Use the combinations result to understand your chances in any combination lottery or random selection scenario.