1. ## combinations

A city has square city blocks formed by a grid of north-south and east-west streets. One automobile route from City Hall to the main firehouse is to go exactly 5 blocks east and 7 blocks north. How many different routes from City Hall to the main firehouse traverse exactly 12 city blocks?

I am not sure how to work this problem.

2. Going exactly 12 blocks is equivalent to saying your motions are either up or right. Here

3. Yeah, but how did you find that there were a total of 210 ways from A to B?

4. You're basically making a "word" out of the following letters:

N N N N N N N E E E E E

(7 "North" and 5 "East" directions.) You can arrange the letters in this "word" any way you like and you'll still get to where you need to go - no matter what order you follow the Ns and Es, you'll eventually go 7 blocks north and 5 blocks east.

So, how many words can you make from these letters?

$\displaystyle \frac {12!}{7! \times 5!}$.

The 12! on top is because there are 12 letters. The 7! and 5! are because 7 of the letters are identical to one another (the Ns) and because 5 of the letters are identical to one another (the Es).

Simplify this expression and you've got the total number of ways.

Note that in the thread that was linked to, the 210 is derived from:

$\displaystyle \frac {10!}{6! \times 4!}$