A town's streets run in a grid. A cyclist starts at one junction and must reach a junction 3 blocks east and 2 blocks north of it, keeping to the streets and never riding south or west. How many different routes can she take?

Every route is 5 blocks long — 3 eastward and 2 northward — and a route is settled once you decide which 2 of those 5 blocks are the northward ones. There are 10 ways to choose 2 things out of 5. A second way to see it is to write on each junction the number of routes reaching it: the bottom row reads 1, 1, 1, 1; the next row 1, 2, 3, 4; the top row 1, 3, 6, 10, since each junction's number is the sum of the numbers just west and just south of it. The traps: 6 multiplies the two block counts, $3\times 2$; 5 adds them; 20 is the count for going 3 blocks east and 3 blocks north, one block too far.