Tuesday, May 5, 2009
solution: streets and avenues
There are 729 ways to travel from the corner of 3rd Ave & 12th St to the corner of 8th Ave and 5th St. In the problem statement, I did not specify that north or west made the numbers go up or down. Correct diagrams could therefore differ, but I did not want to distract from the main riddle, that you must travel five blocks in one direction and seven blocks in the perpendicular direction.
Any valid shortest distance by road will need to be 12 blocks total, five of those being one direction and seven being in the other direction. In my diagram, I need to go east and north. As long as I go five blocks east, seven blocks north, zero blocks west and zero blocks south, I will arrive at the specified location. I can mix up the order...
In other words, I need to travel 12 blocks and 5 of them need to be east (the rest north). Which ones should be east? The answer is a combination, 12C5 = 12!/(5!*7!) = 729 .