The correct answer is 150 and I understand how it needs to be done.

However, my question is: Why can't we solve the question in the following way:

First we take any 3 balls out of 5 and distribute one ball each to the 3 persons. This can be done in C(5,3)*3! ways.

Now we are left with 2 remaining balls. So, we can do either of the following:

i) Give both those balls to one person. This, I believe, can be done in C(3,1) ways Or

ii) Give one ball each to two persons. This, I believe, can be done in C(3,2) ways

So, total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball = C(5,3)*3! *[C(3,1) +C(3,2)] = 360

Why is wrong in the above approach? As I mentioned, the correct answer is 150.