Solution:

To find the number of ways in which we can place the balls in the boxes so that no box remains empty, we will use the principle of inclusion-exclusion.

Let's denote the number of ways in which we can place the balls in the boxes as N. Then, we can write:

N = Total number of ways - Number of ways in which one box is empty + Number of ways in which two boxes are empty - Number of ways in which all three boxes are empty

Total number of ways:

The first ball can be placed in any of the three boxes. Similarly, the second ball can also be placed in any of the three boxes. Continuing this way, we can place all five balls in any of the three boxes. Therefore, the total number of ways is:

3 x 3 x 3 x 3 x 3 = 3^5 = 243

Number of ways in which one box is empty:

There are three ways in which one box can remain empty. Let's assume that box 1 is empty. Then, we can place the five balls in the remaining two boxes in 2^5 = 32 ways. Similarly, we can assume that box 2 or box 3 is empty and find the number of ways. Therefore, the number of ways in which one box is empty is:

3 x 32 = 96

Number of ways in which two boxes are empty:

There are three ways in which we can choose two boxes out of three. Let's assume that boxes 1 and 2 are empty. Then, we can place the five balls in the remaining box in 1 way. Similarly, we can assume different pairs of boxes and find the number of ways. Therefore, the number of ways in which two boxes are empty is:

3 x 1 + 3 x 1 + 3 x 1 = 9

Number of ways in which all three boxes are empty:

There is only 1 way in which all three boxes can remain empty.

Therefore, using the principle of inclusion-exclusion, we can find the number of ways in which we can place the balls in the boxes so that no box remains empty as:

N = 243 - 96 + 9 - 1 = 155

However, the question asks for the number of ways in which we can place the balls in the boxes so that no box remains empty and not the number of ways in which we can place the balls in the boxes with no restrictions. Therefore, we need to subtract the number of ways in which we can place the balls in the boxes with no restrictions but with at least one box empty, which is 3 x 2^4 = 48.

Therefore, the number of ways in which we can place the balls in the boxes so that no box remains empty is:

155 - 48 = 107

Answer: (d) 107.