Combinations & Permutations Calculator

Combinations vs. Permutations

Combinations and permutations are fundamental concepts in combinatorics that help us count how many different ways we can select or arrange elements from a set. The key difference is whether the order of selection matters.

What Are Combinations?

Combinations (nCr) count the number of ways to choose r elements from a set of n elements where order does NOT matter. For example, choosing 3 toppings from a menu of 10. The formula is C(n,r) = n! / (r! x (n-r)!).

What Are Permutations?

Permutations (nPr) count the number of ways to arrange r elements from a set of n elements where order DOES matter. For example, awarding gold, silver, and bronze medals to 3 athletes out of 10 competitors. The formula is P(n,r) = n! / (n-r)!.

Frequently Asked Questions

When do I use combinations vs. permutations?

Use combinations when you are selecting a group and the arrangement does not matter (e.g., lottery numbers, team selection). Use permutations when the arrangement or ranking matters (e.g., passwords, race placements, seat assignments).

What is the maximum value of n supported?

This calculator supports n up to 170 using BigInt precision, which is sufficient for virtually all practical combinatorics problems.