Number of Arrangements of Balls
To find the number of arrangements of at most 88 balls in a row using four different colors, we need to consider all possible combinations of the four colors. Let's break down the problem into smaller parts.

Arrangements with 0 balls
The first possibility is to have 0 balls in the row. This is only one arrangement.

Arrangements with 1 ball
The second possibility is to have only 1 ball in the row. Since there are four different colors, there are four arrangements with 1 ball.

Arrangements with 2 balls
Next, we consider the arrangements with 2 balls. There are two cases to consider:

1. Both balls have the same color: In this case, we have four options (one for each color).
2. The two balls have different colors: In this case, we need to choose two different colors out of the four available. This can be done in C(4,2) = 6 ways. Once the colors are chosen, we can arrange the balls in 2! = 2 ways.

So, the total number of arrangements with 2 balls is 4 + 6 * 2 = 16.

Arrangements with 3 balls
Similarly, we can consider the arrangements with 3 balls. There are two cases to consider:

1. All three balls have the same color: In this case, we have four options (one for each color).
2. Two balls have the same color, while the third ball has a different color: We need to choose two different colors out of the four available, which can be done in C(4,2) = 6 ways. Once the colors are chosen, we can arrange the balls in 3! = 6 ways.

So, the total number of arrangements with 3 balls is 4 + 6 * 6 = 40.

Arrangements with 4 balls
Finally, we consider the arrangements with 4 balls. There are four balls of the same color, so we have four options.

Total Number of Arrangements
To find the total number of arrangements, we add up the arrangements for each case:

1 + 4 + 16 + 40 + 4 = 65

However, the question asks for arrangements of at most 88 balls. Since 65 is less than 88, we can include all the arrangements with 65 balls and add the additional arrangements with 66, 67, ..., 88 balls.

There are 88 - 65 + 1 = 24 additional arrangements possible.

So, the total number of arrangements is 65 + 24 = 89.

The correct answer is option 'B' (87380).