Consider the 6 x 6 square in the figure. Let A1, A2, ..., A49 be the points of intersections (dots in the picture) in some order. We say that A and A are friends if they are adjacent along a row or along a column. Assume that each point A 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 7E(X) is:

The correct answer is 24.

Step 1: Understand the grid points and total number of points
The given image displays a grid formed by a 6 × 6 square, which consists of 6 rows of small squares and 6 columns of small squares.

A grid of 6 × 6 squares has 7 vertical lines and 7 horizontal lines. The points of intersection of these lines are denoted by A₁, A₂, ..., A₄₉.
Total number of intersection points = 7 × 7 = 49.

Step 2: Classify points based on their number of friends (adjacent points along a row or column)
Two points are "friends" if they are adjacent vertically or horizontally. We can categorize the 49 points based on their location on the grid:

1. Corner points:
These are the 4 outer corners of the 7 × 7 grid of points.
Each corner point is adjacent to only 2 other points (1 along the row, 1 along the column).
Number of points with 2 friends = 4.

2. Boundary (edge) points (excluding corners):
Each of the 4 outer edges has 7 points. Excluding the 2 corners on each edge, there are 5 points per edge.
Total boundary points excluding corners = 4 × 5 = 20.
Each of these points is adjacent to 3 other points.
Number of points with 3 friends = 20.

3. Interior points:
These are the inner points of the grid, forming a 5 × 5 grid of interior points.
Total interior points = 5 × 5 = 25.
Each interior point is adjacent to 4 other points (up, down, left, right).
Number of points with 4 friends = 25.

4. Points with 0 or 1 friend:
There are no points with 0 or 1 friend in this grid.
Number of points with 0 friends = 0.
Number of points with 1 friend = 0.

Step 3: Determine the probabilities p_i = P(X = i)
Since each point has an equal chance of being chosen out of the 49 total points, the probabilities for the random variable X representing the number of friends are:

$p_0=P(X=0)=\frac{0}{49}=0$

$p_1=P(X=1)=\frac{0}{49}=0$

$p_2=P(X=2)=\frac{4}{49}$

$p_3=P(X=3)=\frac{20}{49}$

$p_4=P(X=4)=\frac{25}{49}$

Step 4: Calculate the Expected Value E(X)
The expected value E(X) of the random variable X is defined as:

$E\left(X\right)=\sum _{i=0}^{4}i·P(X=i)$

Substituting the values of i and P(X = i):

$E\left(X\right)=0·\left(\frac{0}{49}\right)+1·\left(\frac{0}{49}\right)+2·\left(\frac{4}{49}\right)+3·\left(\frac{20}{49}\right)+4·\left(\frac{25}{49}\right)$

$E\left(X\right)=\frac{8+60+100}{49}=\frac{168}{49}=\frac{24}{7}$

Step 5: Calculate 7E(X)
Now, we find the required value of 7E(X):

$7E\left(X\right)=7·\left(\frac{24}{7}\right)=24$

Thus, the value of 7E(X) is 24.