Lattice Paths to (5,3)

Every path consists of a fixed number of right moves and up moves in some order.

Choose the positions of the up moves among all the moves.

1. 1. Count the moves

   Reaching $(5,3)$ requires $5$ right moves and $3$ up moves, a total of $8$ moves.

   $$\text{total moves}=8$$

2. 2. Choose the order

   Select which $3$ of the $8$ moves are the up moves.

   $$\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$$

Answer: 56