Problem 11 - Algebra and Trigonometry

Question

Towns A, B, C, and D are located in such a way that there are four roads from A to B, five roads from B to C, and six roads from C to D. How many routes are there from town A to town D via towns B and C?

Step-by-Step Solution

Answer

There are 120 different routes from Town A to Town D.

Step 1: Determine Paths from A to B

There are four different roads connecting Town A to Town B. Thus, there are 4 possible ways to travel from Town A to Town B.

Step 2: Calculate Paths from B to C

There are five different roads connecting Town B to Town C. Hence, there are 5 possible ways to travel from Town B to Town C.

Step 3: Determine Paths from C to D

There are six different roads connecting Town C to Town D. This means there are 6 possible ways to travel from Town C to Town D.

Step 4: Compute Total Routes from A to D via B and C

To find the total number of routes from Town A to Town D via Towns B and C, multiply the number of ways between each consecutive pair of towns: \[ 4 \times 5 \times 6 = 120 \]Thus, there are 120 different routes from Town A to Town D.

Key Concepts

Path Counting

Path counting is an essential concept in combinatorics. It is the process of finding the number of distinct paths that can be taken to arrive at a destination when given multiple route options. This is particularly useful in scenarios where you need to navigate between multiple points, choosing different paths at each stage.

Considering our exercise, path counting helps us determine how many different routes exist from Town A to Town D through Towns B and C. By identifying how many roads connect each pair of towns:

• 4 roads from Town A to Town B
• 5 roads from Town B to Town C
• 6 roads from Town C to Town D

You gather all possible paths and multiply them to discover all the potential routes available from start to finish: \( 4 \times 5 \times 6 = 120 \).

This multiplication principle arises from the need to account for every combination of paths across each segment of the journey.

Permutation and Combination

In combinatorics, permutations and combinations are two methods of counting arrangements or selections of items.

Permutations apply when the order of arrangement matters, while combinations are used when order does not matter. For our specific problem of counting paths between towns, while it is technically about arranging paths to create a sequence, the more accurate counting method here mimics combination logic, because we aim to see how different choices combine to form complete routes rather than rearranging them.

Given that different paths between towns can be combined freely, without concern for order between choices (since traveling from A to B doesn't affect the subsequent choice from B to C), it's essential to calculate the different possible paths as part of a multistep process without needing to arrange them differently, hence multiplying choices.

Multistep Process

The multistep process is a sequential approach used when a problem involves multiple stages, where each stage provides several options. Every step must be completed before moving to the next, and the total number of outcomes is the product of the number of choices at each step.

In our scenario, traveling from Town A to Town D is looked at as a three-step process:

• Step 1: Choose one of 4 roads from A to B.
• Step 2: Choose one of 5 roads from B to C.
• Step 3: Choose one of 6 roads from C to D.

Each step represents a decision point with a set number of choices. By calculating the total number of routes via multiplication, we effectively combine each step's choices into a single outcome count.

This approach lets us systematically determine that there are 120 distinct paths from A to D, checking every possible combination of road choices across all steps.