$1 \cdot 69$ Given that the shape of a city is a rectangle, it is divided into many squares by many streets. There are $m$ squares in the north-south direction and $n$ squares in the east-west direction. How many shortest routes are there from one vertex of the rectangle to the opposite vertex?
[Solution] It is evident that the shortest route from one vertex of a rectangle to its opposite vertex consists of $m + n$ segments (each side of a square is considered one segment), among which $m$ segments are north-south oriented, and $n$ segments are east-west oriented. Clearly, each different permutation of the $m$ north-south segments among the $m + n$ segments uniquely determines a route, and the positions of the $m$ north-south segments differ for different routes, i.e., there is a one-to-one correspondence between them. Therefore, the total number of routes is $C_{m+n}^{m}$.