Problem:
The total number of ways in which 5 balls of different colors can be distributed among 3 persons so that each person gets at least one ball is:

a) 75
b) 150
c) 210
d) 243

Solution:

To solve this problem, we can use the concept of distributing identical objects among distinct boxes. Let's break down the solution into smaller steps:

Step 1: Distributing one ball to each person
Since each person needs to get at least one ball, we start by distributing one ball to each person. After this step, we are left with 2 balls and 3 persons.

Step 2: Distributing the remaining 2 balls
Now we need to distribute the remaining 2 balls among the 3 persons. We can think of this as distributing the 2 balls into 3 distinct boxes (each box representing a person), where empty boxes are allowed.

Case 1: One person gets both the remaining balls
In this case, we have 3 choices for selecting the person who will receive both balls. After selecting the person, there is only 1 way to distribute the balls to that person. Therefore, there are 3 ways for this case.

Case 2: Each person gets one ball
In this case, we have 3 choices for selecting the first person to receive a ball, 2 choices for selecting the second person, and only 1 choice for selecting the last person. Therefore, there are 3 x 2 x 1 = 6 ways for this case.

Total number of ways:
Adding up the number of ways from both cases, we get a total of 3 + 6 = 9 ways to distribute the 2 remaining balls.

Total number of ways:
Since step 1 and step 2 are independent, the total number of ways to distribute the balls is the product of the number of ways from both steps. Therefore, the total number of ways is 1 x 9 = 9.

However, we need to consider that the balls are of different colors, which means that the order in which the balls are distributed also matters. Therefore, we need to multiply the total number of ways by the number of ways to arrange the 5 balls among themselves, which is 5!.

Final answer:
The total number of ways is 9 x 5! = 9 x 120 = 1080.

Answer choice:
The correct answer is not among the given options.