Consider the sets $A=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+y^2=25\right\}, B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^2+9 y^2=144\right\}$, $C=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$ and $D=A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

<p>$$\begin{aligned} & A=\left\{(x, y) \in R \times R: x^2+y^2=25\right\}, B=\{(x, y) \in \mathbb{R} \times \\ & \left.\mathbb{R}: x^2+9 y^2=144\right\} \\ & x^2+9 y^2-\left(x^2+y^2\right)=144-25 \\ & \text { Plug in } y^2=\frac{119}{8} \text { into either equation to find } x . \\ & x^2=25-\frac{119}{8} \\ & x^2=\frac{200-119}{8} \\ & x^2=\frac{81}{8} \\ & x= \pm \sqrt{\frac{81}{8}}, y= \pm \sqrt{\frac{119}{8}} \end{aligned}$$</p> <p>Now, $C=\left\{(x, y) \in \mathbb{Z} \times \mathbb{Z}: x^2+y^2 \leq 4\right\}$</p> <p>Valid points are $(-2,0),(-1,-1),(-1,0),(-1,1)$, $(0,-2),(0,-1),(0,0),(0,1),(0,2),(1,-1),(1,0)$, $(1,1)$</p> <p>$\therefore$ Total valid points in $C=13$</p> <p>$\Rightarrow \quad$ There are 4 distinct real points in set $D$</p> <p>$\therefore \quad$ The number of one-one functions from $D$ to $C$</p> <p>$\Rightarrow \quad 13 P_4 \Rightarrow \frac{13!}{(13-4)!}=\frac{13!}{9!}=17160$</p>