A step starting at a point $P$ in the $XY$-plane consists of moving by one unit from $P$ in one of three directions: directly to the right or in the direction of one of the two rays that make the angle of $\pm 120^{\circ}$ with positive $X$-axis. (An opposite move, i.e. to the left/southeast/northeast, is not allowed.) A path consists of a number of such steps, each new step starting where the previous step ended. Points and steps in a path may repeat. Find the number of paths starting at $(1,0)$ and ending at $(2,0)$ that consist of (i) exactly 6 steps (ii) exactly 7 steps.
A step starting at a point $P$ in the $XY$-plane consists of moving by one unit from $P$ in one of three directions: directly to the right or in the direction of one of the two rays that make the angle of $\pm 120^{\circ}$ with positive $X$-axis. (An opposite move, i.e. to the left/southeast/northeast, is not allowed.) A path consists of a number of such steps, each new step starting where the previous step ended. Points and steps in a path may repeat.
Find the number of paths starting at $(1,0)$ and ending at $(2,0)$ that consist of
(i) exactly 6 steps
(ii) exactly 7 steps.