A painter draws 64 equal squares of 1 square inch on a square canvas measuring 64 square inches. She chooses two squares (1 square inch each) randomly and then paints them. What is the probability that two painted squares have a common side?

From the given information, we get that it is $$8\ inch\times\ 8\ inch$$ square grid.

Total ways of selecting 2 squares out of 64 in $$^{64}C_2$$.

Two squares with a common side can be selected in the following ways.

(i) Horizontal Pairs.

In the first row, R1, we can select 7 pairs of squares with a common side.

They are (R1C1,R1C2),  (R1C2,R1C3),.....(R1C7,R1C8).

It applies to other rows as well.

Hence the total number of squares = $$7\times\ 8=56$$

(ii) Vertical Pairs.

In the first column, C1, we can select 7 pairs of squares with a common side.

They are (R1C1,R2C1), (R2C1,R3C1),.....(R7C1,R8C1).

It applies to other columns as well.

Hence the total number of squares = $$7\times\ 8=56$$

The probability of two painted squares having a common side = $$\frac{56+56}{^{64}C_2}$$ = $$\frac{112}{2016}$$.

Option (A) is correct.