Question

Find the number of paths from \(A\) to \(F\) in the following diagram with six letters. A path can only go through letters that are consecutive, either horizontally or vertically, and it goes only to the right or up at each step. $$ \begin{array}{lllll} \text { F } & & & & \\ \text { E } & \text { F } & & & & \\ \text { D } & \text { E } & \text { F } & & & \\ \text { C } & \text { D } & \text { E } & \text { F } & & \\ \text { B } & \text { C } & \text { D } & \text { E } & \text { F } \\ \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { P } \end{array} $$ Prove that a similar path with \(n\) letters has \(2^{n-1}\) paths from the lower left corner to any letter in the rightmost position in a row.

Short answer

2^(n-1) paths exist from the lower left to any rightmost letter with n letters.

Step-by-step solution

Understanding the Problem

In this problem, we need to count the number of valid paths from point \(A\) to point \(F\). The rules state that we can only move either to the right or move up consecutively. The ultimate goal is to prove that for any path of \(n\) letters, there are \(2^{n-1}\) valid paths ending in the rightmost letter.

Analyze Simple Paths

Let's examine simple paths of fewer letters. Starting from one letter ending at itself is considered one path. Consider paths involving just "A" to "F" vertically, where the segments are linear."A to B to C to D to E to F" vertically, there is exactly one path for each transition between letters horizontally or vertically.

Counting Paths from Bottom to Top

We'll use combinatorics to determine the number of paths. Each time we move upwards or to the right, we make one choice out of two possible choices, pressuring us to keep going either upwards or to the right; we'll proceed to detail ways for each corner: "A to B or A to F, or further A to E." Overall path count would be accumulated by multiplying individual choices, shown as 2 choices per move.

Extend the Path Count to n Letters

Extending this to "n" letters: If each path makes exactly one step rightward or upward, the number of possible paths doubles each step because, from each new letter reached, we can choose up or right. Simplified, for \(n\) letters, the number of paths is calculated as \(2^{n-1}\).

Formulating the General Solution for n letters

For a path with \(n\) letters, choosing each next step can be viewed akin to binary decisions — making it clear there are \(2^{n-1}\) total paths when examined substantively through adjacent letters, culminating towards the rightmost node.

Key concepts

Path Counting

In the realm of combinatorics, path counting is a fundamental concept. It involves calculating the number of ways to traverse from one point to another, adhering to specific rules of movement. These paths can vary based on direction and restrictions given in a problem.
For example, in the described problem, paths can only move to the right or vertically upward, which constrains the possible routes one can take from point A to point F. Each movement represents a decision point: move right or go up. This binary choice at each step directly influences the total number of possible paths.
Utilizing simple analysis from smaller segments can help in understanding path dynamics. If you're moving from one letter to another either horizontally or vertically, it's easy to see the straightforward nature of these simple paths. However, when extended to complex routes, the problem transforms into a higher-count path situation involving binary choice outcomes for each step.

Discrete Mathematics

Discrete mathematics is the area of mathematics that deals with countable, distinct elements. It is particularly valued in computer science and combinatorics due to its focus on object separation and arrangement, like the path counting problem presented.
In problems like these, discrete mathematics allows us to abstract and model real-world problems into manageable mathematical formulations. The movement across a grid in this exercise seamlessly translates to discrete steps that involve straightforward calculations, showcased here in the counting paths methodology.
Such methods allow students and mathematicians alike to tackle problems that seem complex by breaking them down into simpler, countable parts. With each movement considered a binary choice, the situation quickly becomes a power set problem, paving the way to apply discrete calculations such as determining paths for an expanding series of steps cumulatively.

Graph Theory

Graph theory is a branch of mathematics that studies relationships, paths, and the connectivity between nodes represented as graphs. This exercise is a great example of practical graph theory application, despite the absence of traditional graph visuals.
When considering points (or letters) as vertices and the connection (or paths) between them as edges, the exercise allows us to visualize the problem using graph theory concepts. The potential to move upward or to the right creates the edges, transforming the letter grid into a directed graph where each step follows established directions only.
By applying graph theory, we can abstract the routes and choices, mapping efficient strategies to arrive at point F from point A. Analyzing how choices and directions define the outcome of reaching a terminal point in this structured graph system forms a cornerstone for understanding involved dynamics in discrete mathematics and combinatorial layouts.