Question

Your family vacation involves a cross-country air flight, a rental car, and a hotel stay in Boston. If you can choose from four major air carriers, five car rental agencies, and three major hotel chains, how many options are available for your vacation accommodations?

Short answer

Answer: There are 60 options available for the vacation accommodations.

Step-by-step solution

Identify the factors for choices

Recognize that there are three independent choices: the air carrier, the car rental agency, and the hotel chain. There are 4 air carriers, 5 car rental agencies, and 3 hotel chains to choose from.

Apply the basic counting principle

The basic counting principle states that when making decisions in a sequence, the total number of possibilities is found by multiplying the number of possibilities for each decision. In this problem, we'll find the product of the number of choices for the air carrier (4), car rental agency (5), and hotel chain (3).

Calculate the number of options

Now calculate the product: 4 air carriers × 5 car rental agencies × 3 hotel chains = 4 × 5 × 3 = 60. There are 60 options available for the vacation accommodations.

Key concepts

Multiplication Principle

The Multiplication Principle is a fundamental concept in combinatorics used to calculate the total number of outcomes for an event consisting of several independent choices. It simplifies the process of determining how many different combinations can be made by choosing one option from each of several categories. For instance, if you have a vacation that involves selecting an air carrier, a rental car, and a hotel stay, the Multiplication Principle will quickly tell you how many unique combinations you can create.

This principle works by multiplying the number of choices available in each category. In our example:

• 4 options for air carriers
• 5 options for car rentals
• 3 options for hotels

The result is: \[4 \times 5 \times 3 = 60\] Thus, there are 60 possible combinations for your vacation, showing how the Multiplication Principle efficiently counts possibilities.

Combinatorics

Combinatorics is the branch of mathematics concerned with counting, arrangement, and combinations of objects. It's the area that helps us solve problems like determining the number of ways to arrange or select items. In the context of your vacation choices, combinatorics is at play to seamlessly blend different options into everyday decision-making.

In these situations, combinatorics provides structured techniques, such as permutations and combinations, to handle complex counting problems. The idea is to systematically combine all possible choices you can make, leveraging tools like the Multiplication Principle. By understanding combinatorics, you gain insight into how diverse arrangements emerge from a set of basic options, just like using airlines, car rentals, and hotels to plan a trip.

It's not just for vacations—combinatorics applies to many scenarios, from scheduling to creating passwords.

Probability

Probability deals with the likelihood or chance of different outcomes occurring in situations of uncertainty. While combinatorics counts possible outcomes, probability assesses how likely each one is to happen. In the case of your vacation planning, understanding probability would involve determining if some choices are more favorable or likely than others.

When applying probability, the basic formula involves dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you were interested in the probability of choosing a specific airline out of the four available options, you'd calculate: \[\text{Probability} = \frac{1}{4}\] This is because there is one favorable outcome (choosing your specific airline) and four total choices.

Incorporating probability with the Counting Principle provides an even deeper understanding of how likely different combinations are, adding a layer of analysis to simple enumeration.