1977 AHSME Problems/Problem 20
Problem
For how many paths consisting of a sequence of horizontal and/or vertical line segments, with each segment connecting a pair of adjacent letters in the diagram above, is the word CONTEST spelled out as the path is traversed from beginning to end?
Solution 1
The path will either start on the left side, start on the right side, or go down the middle. Say it starts on the left side. Instead of counting it starting from the 
, let's start at the T at the end and work backwards. There is 
option to choose the 
. Then, there are 
options to choose the 
, 
, , 
, 
, and , for a total of 
possibilities. But we need to subtract the case where it does down the middle, which makes it 
possibilities. The right side also has possibilities by symmetry. There is way for it to go down the middle, for a total of 
ways, so the answer is 
.
