JEE Advanced 2023 — Paper 2 — Question 37

Consider the $6 \times 6$ square in the figure. Let $A_{1}, A_{2}, \ldots . ., A_{49}$ be the points of intersections (dots in the picture) in some order. We say that $A_{\mathrm{i}}$ and $A_{\mathrm{j}}$ are friends if they are adjacent along a row or along a column. Assume that each point $A_{\mathrm{i}}$ has an equal chance of being chosen.

Let $p_{i}$ be the probability that a randomly chosen point has $i$ many friends, $i=0,1,2,3,4$. Let $X$ be a random variable such that for $i=0,1,2,3,4$, the probability $P(X=i)=p_{i}$. Then the value of $7 E(X)$ is