Clever counting » Routes in a grid

Example

How many routes are there from A to C via B, without detours?
    Grid of 5 grid squares high and 8 grid squares wide and points A(0,0), B(4,2) and C(8,5)

Answer:
A route from A to B without detours consist of 6 steps.
For example ENENEE. 2 steps North and 4 steps East.
This has to apply to every route from A to B, so how many routes are possible that consist of 2N and 4E?
You can calculate that is:
bracket open6
2
bracket close = 6 · 52 · 1 = 6 nCr 2 = 15

In the same way you can calculate the number of routes from B to C:
bracket open7
3
bracket close = 7 · 6 · 53 · 2 · 1 = 7 nCr 3 = 35

So the total number of routes is: 15 × 35 = 525