A secretary works in a building located nine blocks east and...

A secretary works in a building located nine blocks east and eight blocks north of his home. Every day he walks 17 blocks to work. (See the map that follows.) (a) How many different routes are possible for him? (b) How many different routes are possible if the one block in the easterly direction, which begins four blocks east and three blocks north of his home, is under water (and he can't swim)? (Hint Count the routes that use the block under water.)

A secretary works in a building located nine blocks east and eight blocks north of his home. Every day he walks 17 blocks to work. (See the map that follows.)
(a) How many different routes are possible for him?
(b) How many different routes are possible if the one block in the easterly direction, which begins four blocks east and three blocks north of his home, is under water (and he can't swim)? (Hint Count the routes that use the block under water.)

Key Concepts

Multiplication Principle

The multiplication principle is a basic rule in combinatorics stating that if a task can be divided into independent sub-tasks, the total number of ways to complete the overall task is the product of the numbers of ways to perform each sub-task. In grid path problems, this is often applied when a route is segmented into parts, for example, finding the number of paths via an intermediate point.

Obstacle Avoidance in Grid Paths

Obstacle avoidance in grid path problems involves adjusting the count of available paths when certain segments or points are blocked. This typically requires calculating the number of paths that encounter the obstacle (by splitting the journey into segments before and after the obstacle) and then subtracting them from the total count of unrestricted paths.

Lattice Paths

Lattice paths refer to routes on a grid where movement is restricted to certain directions (commonly east and north). This concept is fundamental in problems where one must count the number of ways to travel from one point to another on a grid, given that steps can only be taken in specific, allowed directions.

Binomial Coefficients

Binomial coefficients are used to count the number of ways to choose a subset of moves when the total number of moves is fixed. In the context of grid paths, if a path is composed of a certain number of moves in two different directions, the number of distinct paths is given by the binomial coefficient determining how to arrange these moves.

Transcript

Please give Ace some feedback