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 square is divided into three equal parts horizontally to create [Figure 1], and divided into three equal parts vertically to create [Figure 2]. [Figure 1] and [Figure 2] are alternately attached repeatedly to create the following figure. As shown in the figure, let A be the upper left vertex of the first attached [Figure 1], and let $\mathrm { B } _ { n }$ be the lower right vertex of the figure created by attaching a total of $n$ figures (combining the number of [Figure 1] and [Figure 2]).
When $a _ { n }$ is the number of shortest paths from vertex A to vertex $\mathrm { B } _ { n }$ along the lines, what is the value of $a _ { 3 } + a _ { 7 }$? [4 points]
(1) 26
(2) 28
(3) 30
(4) 32
(5) 34

There is a walking path in a rectangular lawn. As shown in the figure, this walking path consists of 8 circles with equal radii that are externally tangent to each other.
Starting from point A and arriving at point B along the walking path by the shortest distance, find the number of possible routes. (Note: The points marked on the circles represent the points of tangency between the circles and the rectangle or between the circles.) [4 points]

We assume that $d = 2$ and that the distribution of $X$ is given by $$P ( X = ( 0,1 ) ) = P ( X = ( 0 , - 1 ) ) = P ( X = ( 1,0 ) ) = P ( X = ( - 1,0 ) ) = \frac { 1 } { 4 }$$ Let $n \in \mathbb{N}$. Establish the equality $$P \left( S _ { 2 n } = 0 _ { 2 } \right) = \left( \frac { \binom { 2 n } { n } } { 4 ^ { n } } \right) ^ { 2 }$$

A person $X$ standing at a point $P$ on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps $X$ comes to the original position $P$. Then the number of distinct paths that $X$ can take is
(a) 196
(b) 256
(c) 344
(d) 400

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices $(0, 0)$, $(0, 41)$ and $(41, 0)$, is:
(1) 820
(2) 780
(3) 901
(4) 861