Question

Your family vacation involves a cross-country air flight, a rental car, and a hotel stay in Vancouver. 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: 60 options.

Step-by-step solution

Identify the number of choices for each element

In this case, there are three elements (air carriers, car rental agencies, and hotel chains). We have four major air carriers, five car rental agencies, and three major hotel chains. So there are 4 choices for air carriers, 5 choices for car rental agencies, and 3 choices for hotel chains.

Apply the counting principle

According to the counting principle, we need to multiply the choices of each element to get the total number of options. So, multiply the number of air carriers (4), car rental agencies (5), and hotel chains (3). Total options = (number of air carriers choices) * (number of car rental agencies choices) * (number of hotel chains choices)

Calculate the total number of options

Now, multiply 4 * 5 * 3 to find the total number of options. Total options = 4 * 5 * 3 = 60 Therefore, there are 60 options available for vacation accommodations.

Key concepts

Probability

Probability is a way to measure how likely it is for an event to happen. It ranges from 0 to 1, where 0 means the event will not happen, and 1 means it definitely will.
In the context of our vacation planning problem, probability helps to understand the likelihood of choosing a specific combination of air carrier, car rental, and hotel.

• If you wanted to find the probability of picking a specific combination (let's say one air carrier, one car rental, and one hotel), you would calculate the likelihood of choosing each of those options separately and then multiply them together.
• For example, if each choice is equally likely, and you have 60 total combinations, the probability of picking any one combination would be \({1}/{60}\).

Understanding probability shines a light on how preferences and constraints might influence your decision-making.

Combinatorics

Combinatorics is the area of mathematics focusing on counting, arrangements, and combinations. It's about figuring out how many different ways you can do something.
When planning a trip, like in our exercise, combinatorics is used to find the total number of possible vacation packages by combining various elements such as air carriers, car rental agencies, and hotel chains.

• For instance, you can think of each choice (airlines, cars, hotels) as a separate category where you choose one option from each category.
• By listing all possible combinations, you ensure that you have covered every potential arrangement before making a decision.

This approach aids in comprehensive planning, ensuring no option is overlooked.

Multiplication Principle

The multiplication principle is a foundational idea in combinatorics and probability, used to find the number of possible outcomes for a sequence of choices. It's about making decisions in stages, each independent of the others.
In our example, each choice (air carrier, car rental, hotel) is independent, meaning the choice of air carrier doesn’t affect the choice of hotel.

• To find the overall number of combinations, you multiply the number of choices for each individual decision.
• So, multiplying 4 air carriers by 5 car rental options by 3 hotels gives \(4 \times 5 \times 3 = 60\) total choices.

This principle provides a quick way to calculate the total number of possible combinations efficiently, without listing them all individually.