High School Permutations and Combinations Calculator
Calculate the number of ways to choose r items from a set of n.
Permutations (Order Matters)
Arrangements
Combinations (Order Doesn’t Matter)
Groups
Comparison Visualization
Understanding the Permutations and Combinations Calculator
Counting might seem simple, but when dealing with large sets, calculating the number of possible outcomes requires specific mathematical formulas. The two main ways to count groupings are Permutations and Combinations. The critical difference between them is whether the order of the selected items matters.
Permutations (Order Matters)
A permutation is an arrangement of items in a specific order.
Analogy: A combination lock (which should really be called a permutation lock!). The code “1-2-3” is different from “3-2-1”.
Formula: P(n, r) = n! / (n – r)!
Combinations (Order Doesn’t Matter)
A combination is a selection of items where the order is irrelevant.
Analogy: A fruit salad. A bowl with “Apple, Banana, Grapes” is the exact same meal as “Grapes, Apple, Banana”.
Formula: C(n, r) = n! / [r! (n – r)!]
Why use a Permutations and Combinations Calculator?
As n grows, the factorials involved grow explosively. Calculating 15! by hand is impractical. This tool handles the large arithmetic instantly and, uniquely, helps you visualize the disparity between the two concepts. Notice how Permutations are always greater than or equal to Combinations because every single combination can be arranged in multiple orders.
FAQ
Yes, this tool is completely free for students and teachers.
This is “n factorial”. It means multiplying a series of descending natural numbers. For example, 4! = 4 × 3 × 2 × 1 = 24.
It can calculate standard JS floating point limits (up to 170!). For extremely large inputs, results may be displayed in scientific notation.
Because nCr divides the nPr result by r! to remove the duplicate arrangements of the same items.