Understanding Permutations and Combinations
Permutations and combinations are fundamental concepts in combinatorics used to count the number of ways to arrange or select items from a set. They are essential in probability theory, statistics, and many real-world applications such as lottery calculations, password generation, and experimental design.
Permutations (nPr)
A permutation counts the number of ways to arrange r items from a set of n items where order matters.
nPr = n! / (n - r)!
Example: How many ways can 3 runners finish in 1st, 2nd, and 3rd from 10 competitors? 10P3 = 10! / 7! = 720
Combinations (nCr)
A combination counts the number of ways to choose r items from n items where order does not matter.
nCr = n! / ((n - r)! × r!)
Example: How many ways can you choose 3 members from a group of 10? 10C3 = 120
Basic Probability
The probability of an event A is the ratio of favorable outcomes to total possible outcomes. It always falls between 0 (impossible) and 1 (certain).
P(A) = favorable outcomes / total outcomes
Frequently Asked Questions
When should I use permutations vs. combinations?
Use permutations when the arrangement order matters (e.g., rankings, passwords). Use combinations when you only care about which items are selected, not their order (e.g., choosing team members, lottery numbers).
What is the maximum value of n this calculator supports?
This calculator handles values up to n = 170, since 171! exceeds JavaScript's maximum number range. For most practical purposes, this range is more than sufficient.