Lattice points are points on a rectangular coordinate plane having both $x$ - and $y$-coordinates being integers. A moving point $P$ is initially located at $(0,0)$. It moves 1 unit along the coordinate lines (in either directions) in a single step.
G10.1 If $P$ moves 1 step then $P$ can reach $a$ different lattice points, find $a$.

G10.2 If $P$ moves not more than 2 steps then $P$ can reach $b$ different lattice points, find $b$.

G10.3 If $P$ moves 3 steps then $P$ can reach $c$ different lattice points, find $c$.

G10.4If $d$ is the probability that $P$ lies on the straight line $x+y=9$ when $P$ advances 9 steps, find $d$.
$\begin{array}{l}(1,0),(-1,0),(0,1),(0,-1) \\ a=4 \\ (1,0),(-1,0),(0,1),(0,-1) \\ (1,1),(1,(1,-1),(-1,1),(-1,-1) \\ (2,0),(-2,0),(0,2),(0,-2),(0,0) \\ b=13 \\ (1,0),(-1,0),(0,1),(0,-1),(3,0),(2,1),(1,2),(0,3),(-1,2),(-2,1),(-3,0),(-2,-1), \\ (-1,-2),(0,-3),(1,-2),(2,-1) ; c=4+12=16 \\ \text { Total number of outcomes }=4+12+20+28+36=100 \\ \text { Favourable outcomes }=\{(0,9),(1,8),(2,7), \ldots,(9,0)\}, \text { number }=10 \\ \text { Probability }=\frac{1}{10}\end{array}$