Combination Calculator

About Combination Calculator

A combination is a mathematical technique that determines the number of possible arrangements in a collection of items where the order of selection does not matter. In combinations, you can select the items in any order.

Combination Formula

The formula to determine the number of possible combinations when selecting r items from a set of n items without repetition is:

C(n,r) = n! / (r!(n-r)!)

Where:

• n = total number of items in the set
• r = number of items to select
• ! = factorial (e.g., 4! = 4 × 3 × 2 × 1 = 24)

With Repetition:

When repetition is allowed, the formula becomes:

C(n+r-1, r) = (n+r-1)! / (r!(n-1)!)

Combination vs Permutation

Combinations and permutations are similar concepts, but with one key difference:

• Combination: Order doesn't matter (e.g., AB is the same as BA)
• Permutation: Order matters (e.g., AB is different from BA)

Examples of Combinations

Example 1: How many different committees of 3 people can be formed from 10 people?

C(10,3) = 10! / (3! × 7!) = 120

Example 2: How many different ice cream combinations can you make with 5 flavors if you can choose 2 scoops (repetition allowed)?

C(5+2-1, 2) = 6! / (2! × 4!) = 15

Applications of Combinations

Combinations are used in various fields including:

• Probability theory
• Statistics
• Computer science (algorithm design)
• Game theory
• Genetics (combinations of genes)
• Lottery calculations