How to Use the Combination Calculator

Select the type of combination calculation, enter your values, and click Calculate. The calculator provides three different modes for various combinatorial problems.

Types of Combinations

Standard Combination (nCr)

The number of ways to choose r items from n distinct items where order does not matter. Also called binomial coefficient. Formula: C(n,r) = n! / (r! × (n-r)!). For example, C(5,3) = 10 represents the 10 ways to choose 3 items from 5.

Multiset Combination (With Repetition)

When items can be repeated, use the multiset formula. If you have n types of items and want to choose r total items, the formula is C(n+r-1, r). For example, choosing 3 fruits from 5 types (apple, orange, banana, grape, mango) where you can pick the same fruit multiple times.

Pascal's Triangle Row

Generate a specific row of Pascal's triangle. Each row n contains the binomial coefficients C(n,0), C(n,1), ..., C(n,n). The sum of row n equals 2^n. Pascal's triangle has many fascinating properties in mathematics.

Combinations vs Permutations

In combinations, order does not matter: choosing ABC is the same as choosing CBA. In permutations, order matters: ABC is different from CBA. Combinations are always less than or equal to permutations for the same n and r values. Specifically, C(n,r) = P(n,r) / r!

Common Applications

Lottery probability calculations (how many ticket combinations exist)
Committee selection problems
Hand combinations in card games
Statistical sampling methods
Binomial probability distributions

Pascal's Triangle Properties

Each number is the sum of the two numbers above it
The sum of row n equals 2^n
The triangle is symmetric (C(n,r) = C(n,n-r))
Rows contain binomial coefficients used in polynomial expansion