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In combinatorics, a combination is a selection of items from a larger set, where the order of selection does not matter. Combinations are often used to count the number of ways to choose a subset of objects from a larger set.

Formula

The number of combinations of $$k$$ items chosen from a set of $$n$$ items (denoted as $$C\left(n,k\right)$$ or $$n\text{[PARSE ERROR: Undefined("Command("choose")")]}k$$) is calculated using the combination formula:

where $$n!$$ represents the factorial of $$n$$, defined as the product of all positive integers less than or equal to $$n$$.

Properties

• The number of combinations is always a non-negative integer.
• Combinations are unordered, meaning that selecting the same set of items in a different order does not create a new combination.
• The number of combinations is often used in probability calculations and counting problems.

Example

Suppose we have a set of 5 letters: A, B, C, D, and E. We want to choose 3 letters from this set without regard to the order of selection. The number of possible combinations is:

Therefore, there are 10 different combinations of 3 letters that can be chosen from the set {A, B, C, D, E}.

Combinations are fundamental in combinatorial mathematics and have numerous applications in various fields, including statistics, computer science, and cryptography.