4A. In how many ways can the king on an $8 \times 8$ chessboard move from the bottom left corner to the top right corner, moving only right and forward?
Solution. For the king to reach from the bottom left corner to the top right corner, he needs to move seven times to the right and seven times forward. Let's denote the movement to the right as R, and the movement forward as F. The problem boils down to the following: in how many ways can seven R's and seven F's be arranged. This is a case of permutations of 14 elements where 7 are repeated and the other 7 are repeated. The number is $\frac{14!}{7!7!}=3432$.