JEE Main 2025 Apr 4 Shift 1, Mathematics Q2: Permutation

Step 1 - find $|D|$. $A$ is a circle and $B$ is an ellipse. Eliminating $y^2$ from $x^2+y^2=25$ and $x^2+9y^2=144$ gives $8x^2=81$, so $x=\pm \frac{9}{2\sqrt{2}}$ and correspondingly $y=\pm \frac{\sqrt{119}}{2\sqrt{2}}$. All four sign combinations are valid, so $D$ has 4 elements.

Step 2 - find $|C|$. $C$ consists of integer lattice points inside or on the disc of radius 2. Listing them: $(0,0),(\pm 1,0),(0,\pm 1),(\pm 2,0),(0,\pm 2),(\pm 1,\pm 1)$ - that is $1+4+4+4=13$ points.

Step 3 - count one-one maps. Choosing distinct images for 4 elements out of 13: $13\times 12\times 11\times 10=17160$.