Suppose you have a standard 8by8 chessboard and you place a

Suppose you have a standard (8-by-8) chessboard, and you place a marker on the bottom left square, which we\'ll call (1, 1). Your marker can now take single moves: either the marker can move one square to the right, or one square upward at any step. Thus, it will take 14 moves to get the marker to the upper right corner, whichwe\'ll call (8, 8).

How many distinct 14-move paths are there by which the marker can go from the bottom left corner to the upper right corner?

Solution

You have to move exactly seven times to the right and exactly seven times up.

These can be done in any order, However a move can be written as a sequence of U\'s and R\'s where U\'s represents a movement upward and R represents rightward movement.

So one move could be
URURURURURURUR.
So number of ways the first move can be taken is 14 ! /(7!)^2.