Problem 78 - College Algebra

Question

What is a combination?

Step-by-Step Solution

Answer

A combination is a method of selecting items from a larger set where the order does not matter. The number of combinations can be found using the formula \( C(n, r) = \frac{n!}{r!(n-r)!} \).

Step 1: Understanding Combination

A combination is a way of selecting items from a larger set. What makes combinations unique is that the order of the items selected does not matter. For example, if you have a set of three items - A, B, and C. A combination of two items could be AB, AC, or BC. In combinations, AB and BA are considered the same because the order does not matter.

Step 2: Combination Formula

The formula to calculate combinations is given by: \( C(n, r) = \frac{n!}{r!(n-r)!} \) \nWhere: \n \( C(n, r) \) is the number of combinations. \n \( n \) is the total number of items. \n \( r \) is the number of items to select. \n\( ! \) means factorial, which is the product of all positive integers up to that number. For example, \( 5! = 5 × 4 × 3 × 2 × 1 \)

Step 3: Using the Combination Formula

To use the formula, you simply substitute your total number of items for \( n \) and the number of items to select for \( r \). For example, if you have a set of 5 items (n=5) and you want to find out how many ways you can select 3 items (r=3), you would calculate \( C(5, 3) = \frac{5!}{3!(5-3)!} \)