Problem 64 - College Algebra

Question

What is a combination?

Step-by-Step Solution

Answer

A combination is a mathematical concept used for finding the number of ways of selecting items from a larger set. The order of selection does not matter in combinations. The mathematical formula for it is \( C(n, r) = \frac{n!}{r!(n-r)!} \). For a set of three items {a, b, c}, there are three ways to choose two items: {a, b}, {a, c}, and {b, c}.

Step 1: Definition

A combination is a selection of items from a larger set where the order of selection does not matter. It's an arrangement of items into subsets, not considering the order. In other words, if the order doesn't matter, it is a combination.

Step 2: Mathematical Representation

Combinations can be represented mathematically as \( C(n, r) = \frac{n!}{r!(n-r)!} \). In this formula, \( n \) refers to the total number of items in the set, \( r \) is the number of items to choose, and \( ! \) means factorial, which is the product of an integer and all the integers below it down to 1.

Step 3: Notation

The notation \( C(n, r) \) or \( _nC_r \) is used to represent a combination. It's read as 'n choose r', indicating how many ways you can choose r items from a set of n.

Step 4: Examples

For example, if you have a set of three items {a,b,c} and you want to choose two items, you will have 3 combinations: {a,b}, {a,c}, and {b,c}.