Problem 5 - Algebra for College Students

Question

A parcel delivery truck can take any one of 3 different roads from Clarksville to Leesville and any one of 4 different roads from Leesville to Jonesboro. How many different routes are available from Clarksville to Jonesboro?

Step-by-Step Solution

Answer

There are 12 different routes available.

Step 1 - Understanding the Problem

Identify the number of choices available at each stage of the journey. There are 3 different roads from Clarksville to Leesville and 4 different roads from Leesville to Jonesboro.

Step 2 - Applying the Counting Principle

Multiply the number of choices at each stage to find the total number of different routes. This uses the fundamental counting principle which states that if there are 'a' ways to do something and 'b' ways to do another thing, then there are a x b ways of doing both.

Step 3 - Calculate the Total Number of Routes

Multiply the number of roads from Clarksville to Leesville (3) by the number of roads from Leesville to Jonesboro (4): Number of routes = 3 * 4 = 12.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics focusing on counting, arrangement, and combination of objects.
It is essential for solving problems where you need to figure out the number of possible ways to do something.
In our example, combinatorics helps us determine how many different routes the delivery truck can take from Clarksville to Jonesboro.
By understanding how to count these combinations correctly, we can simplify complex problems with multiple steps.
Often, combinatorics involves the use of the __fundamental counting principle__, which we'll explore further in the next section.

Multiplication Principle

The multiplication principle, a fundamental concept in combinatorics, makes counting combinations easier.
It states that if there are 'a' ways to do one task and 'b' ways to do a second task, the total number of ways to do both tasks is the product: \( a \times b \).
In our exercise, we have two tasks: choosing a road from Clarksville to Leesville and choosing a road from Leesville to Jonesboro.
First, there are 3 possible roads for the truck to travel from Clarksville to Leesville.
Second, there are 4 possible roads available from Leesville to Jonesboro.
Therefore, using the multiplication principle, we multiply these numbers: \( 3 \times 4 = 12 \).
The total number of different routes from Clarksville to Jonesboro is 12.

Route Calculation

Route calculation combines our understanding of combinatorics and the multiplication principle to solve real-world problems.
In this exercise, we calculated the different possible routes a delivery truck can take using a step-by-step approach.
First, we identified the choices at each stage: 3 roads to Leesville, and 4 roads to Jonesboro.
Then, we applied the multiplication principle to find the total number of routes: \( 3 \times 4 = 12 \).
This systematic approach can be applied to various similar problems, such as determining the number of outfit combinations or scheduling multiple tasks.
By learning to break down the problem into steps and applying fundamental principles, we can solve complex problems with ease.