As we can see, it is possible to paint 25 different horizontal rectangles. In a similar way we conclude that on rectangle it is possible to paint horizontal rectangles and different vertical rectangles. We have possible combinations in total.
On square it is possible to paint 4 different intersections of vertical and horizontal rectangles. We can consider different squares on the rectangle. Thus, we obtain different combination, when two rectangles intersect each other. I.e., there will be cells painted twice.
Thus, the probability that one cell is painted twice is:
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