1. Interpret the grid

A $6 \times 6$ square divided into unit squares has intersection points arranged in a $7 \times 7$ grid.

So total number of points is $49.$

Each point can have friends only in the four possible directions: left, right, up, down, provided such adjacent points exist.

Thus the number of friends of a point depends on its position.

2. Classify the points

There are three types of points:

(i) Corner points

There are $4$ corner points. Each corner has exactly $2$ friends.

(ii) Edge points (excluding corners)

Each side has $7$ points, so excluding the $2$ corners leaves $5$ edge points per side. Thus total edge-but-not-corner points: $4 \times 5 = 20.$ Each such point has exactly $3$ friends.

(iii) Interior points

These are the points not on the boundary. Count: $49 - 4 - 20 = 25.$ Each interior point has exactly $4$ friends.

3. Find the probabilities $p_i$

Since each of the $49$ points is equally likely:

• $p_0 = 0$
• $p_1 = 0$
• $p_2 = \dfrac{4}{49}$
• $p_3 = \dfrac{20}{49}$
• $p_4 = \dfrac{25}{49}$

4. Compute the expectation $E(X)$

By definition, $E(X)=\sum_{i=0}^4 i\,p_i.$ So, $E(X)=0\cdot 0 + 1\cdot 0 + 2\cdot \frac{4}{49} + 3\cdot \frac{20}{49} + 4\cdot \frac{25}{49}.$

Now simplify: $E(X)=\frac{8+60+100}{49}=\frac{168}{49}=\frac{24}{7}.$

5. Find $7E(X)$

$7E(X)=7\cdot \frac{24}{7}=24.$

6. Compare with stored answer

Derived answer is $24$, which matches the stored correct answer.