QJEEAdv23Ma6ffa: JEE Advanced 2023 Mathematics - Class 12 JEEAdv23Ma6ffa Solution: Let be the probability that a randomly chosen point has many…

Number of points having 1 friend $=0$

Number of points having 2 friends $=4$

Number of points having 3 friends $=5 \times 4=20$

Number of points having 4 friends $=49-24=25$

$\mathrm{P}_{\mathrm{i}}=$ Probability that randomly selected points has friends

$\mathrm{P}_0=0$ (0 friends)

$\mathrm{P}_1=0$ (exactly 1 friends)

$\mathrm{P}_2=\frac{{ }^4 \mathrm{C}_1}{{ }^{49} \mathrm{C}_1}=\frac{4}{9}$ (exactly 2 friends)

$\mathrm{P}_3=\frac{{ }^{20} \mathrm{C}_1}{{ }^{49} \mathrm{C}_1}=\frac{20}{49}$ (exactly 3 friends)

$\mathrm{P}_4=\frac{{ }^{25} \mathrm{C}_1}{{ }^{49} \mathrm{C}_1}=\frac{25}{49}$ (exactly 4 friends)