Use the formula for nCr to solve Exercises 49–56. To win at LOTTO in the state of Florida, one must correctly select 6 numbers from a collection of 53 numbers (1 through 53). The order in which the selection is made does not matter. How many different selections are possible?

Key Concepts

Combinations (nCr)

Combinations, denoted as nCr, represent the number of ways to choose r items from a set of n items without regard to the order of selection. The formula for combinations is nCr = n! / (r!(n - r)!), where '!' denotes factorial, the product of all positive integers up to that number. This concept is essential for solving problems where the arrangement of selected items is irrelevant.

Factorial

A factorial, denoted as n!, is the product of all positive integers from 1 to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are crucial in combinatorial calculations, as they help determine the total number of arrangements or selections possible within a set. Understanding how to compute factorials is necessary for applying the combinations formula.

Counting Principles

Counting principles, such as the fundamental counting principle, provide a systematic way to count the number of outcomes in a scenario. In the context of combinations, it helps to understand how to systematically select items from a larger set. This principle is foundational in probability and combinatorics, allowing for the calculation of possible selections in various contexts, such as lottery games.