| Suppose the streets of Alma are laid out in a rectangular grid with my home located at (0,0) and Alma College located at (10,10). Each day I travel the 20 blocks from my home to school stopping briefly at Java Joe’s, a coffee shop located at (3,5). In how many different ways can I make this 20-block trip? An example of one such trip is given. |
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Solution by Andrew Le , Troy Athens High School
First, split the problem into two different parts:
1. the trip from home to Java Joe's and
2. the trip from Java Joe's to Alma College.
Since the trip needs to be a 20-block trip, then you can only go right or up. For the first part, there are a total of eight steps that need to be taken to get to Java Joe's. Three steps need to be to the right, and five steps need to be up. Since it doesn't matter when you go up or right as long as you go right 3 times and up 5 times, you can compute the number of possibilities using 
(8 choose 3) or 
(8 choose 5). So, the number of ways to get from home to Java Joe's 
.
You can use the same concept with the second part. Since it doesn't matter when you go up or right as long as you go right 7 times and up 5 times, you can compute the number of possibilities from Java Joe's to Alma College using 
or 
. So, the number of ways to get from Java Joe's to Alma College 
.
Now, you multiply the two numbers because for every path from home to Java Joe's there are 792 different paths from Java Joe's to Alma College. So, the total number of paths from home to Alma College is 
.
Note: The notation, 
, refers to the number of combinations of r items selected from n different items, and we compute this number as follows: 
.
Also solved by: Nick Wasylyshyn (Canton H. S.); Niels Westerink (Vestaburg H.S.); Jim Ross (Warren Mott H.S.); Alex Robinson (Lansing Catholic H.S.); Frank Ruizhi Hua (Milan H.S.); Phil Dirkse (Jenison H.S .); Caroline Bjerregaard, Amanda Gade, Whitney Herman, Brian Siebers (Forest Hills Northern H.S.); Adam Booth, Allison Bryan, Max Dixon, Rebecca Hammond, Drew Reyelts, Nicholas Sayles, Marc Walker (Forest Hills Eastern H.S.); Karrie Juengel, Theresa Laubrick (Linden H.S.); Manish Patel (Saginaw Heritage H.S.); Allen Farmer (Flushing H.S.); and Elizabeth Lyons (Alamo Heights H.S., San Antonio, TX) .