Probability

# Rectangular Grid Walk - No Restrictions

Howard lives in Pleasantville on Park St and First St. His favorite grocery store is located a couple blocks away on Front and Fourth. Main St is the only street between Park St and Front St. How many routes could he take to drive from his house to the store, provided he wants to get there as quickly as possible?

Cindy is playing a video game and finds herself traveling from one location to another. Fairly quickly, she notices that she is only pressing the up and right buttons, each of which move her character one space. If Cindy is trying to move a total of $8$ spaces to the right and $2$ spaces up, in how many different ways can she get to her destination?

The Andromedan Trade Goods Association requires the following of its member worlds, whose locations can be plotted as a $100 \times 100$ grid.

• Each time a world receives a trade good, it must send that trade good to one of the worlds immediately to the right of or above it.
• Another trade good must be sent to the other world.

If the world in the bottom left sends out a trade good to each of the two worlds next to it, how many trade goods will the world $12$ to the right and $3$ above the initial world receive?

How many ways can Little Red Riding Hood move from her home at $(2, \, 3)$ to Grandmother's house at $(9, \, 2)$ if she must first visit the river along the $x$-axis and each move she takes must be one unit to the right, up, or down, with all down moves taking place before any up moves?

Note: She can pass through her Grandmother's house on the way to the river, but she then needs to go back and visit it.

How many ways can a particle move from $(2, \, 3)$ to $(7, \, 5)$ if each move must be $1$ unit up or to the right?

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