Chess Antics

Imagine that you're an ant and are standing at the top left starting point of the rook, facing towards the down right, the destination. Now, with your frame of reference, it'll look something like this:- Chessboard

Now, we need to go from $(0,0)$ to the lattice point $(7,7)$ . So, we need to go up the axis $7$ times and towards the right $7$ times. Let's denote them by $x$ and $y$ respectively.

We need $7$ $x's$ and $7$ $y's$ , no matter the order, we'll reach the same place.

Something like $x y x x y y y x y x y x y x$ will do.

So we need a total of $14$ things, divided in $2$ groups of $7$ same things each.

Now it's easy. The number of ways to order $14$ things, of which $7$ are of one kind (x) and $7$ are of another king (y) is given by:-

$\frac{14!}{7! \times 7!}= \boxed{3432}$

Hope everything is explained. I know that the imagination was unnecessary, but it makes the problem and the solution more interesting and easy to understand. -Satvik

The rook will have to move 7 steps right and 7 steps down. The number of such paths is the number of combinations of 14 steps, 7 of which are down and the other 7 of which are right. That`s 14C7= 3432 ways.