Well, let's see here. According to my calculations and my impeccable sense of humor, the correct answer is... (drumroll, please)... c) 3.75mg!
Why, you ask? Well, because when the two masses are hanging vertically and at rest, the tension in the string on one side is equal to the weight of that mass (3m) and on the other side it is equal to the weight of the other mass (5m). When the system is released, the heavier mass will start moving downwards, causing the pulley to rotate. This rotation will cause the tension in the string to change.
Now, if we take a closer look at the forces involved, we can observe that the tension in the string on the heavier side must be greater than the tension on the lighter side. This is because the heavier mass wants to accelerate downwards, so the tension must be strong enough to counteract that. As a result, the pulley will experience a net force in the direction of the lighter mass.
So, the magnitude of the resultant force exerted by the string on the pulley can be calculated by finding the difference between the tensions on each side. The tension on the heavier side is 5mg, and the tension on the lighter side is 3mg. Subtracting the two, we get 5mg - 3mg = 2mg.
But hold on, we're not done yet! Since the pulley is experiencing a net force in the direction of the lighter mass, we need to double the result to take into account both sides of the pulley. Therefore, the correct answer is 2mg * 2 = 4mg!
Oh wait, that's not right, is it? I apologize for the confusion, my dear friend. It seems I have stumbled upon a mathematical blunder. The correct answer is indeed e) 2mg. I'm sorry for the mix-up, but I hope my attempt at humor brought a smile to your face nonetheless!