I feel ya! (an idiom...just in case it is misunderstood)

I don't think you are interpreting the problem correctly. the problem never said you had to keep picking until you ended up with the "combination". It simply says, if you were to pick 3 balls (with replacement) what is the probability that you will pick a red, then a green, then a blue.

The probability of picking a red ball is:

that is, you can choose 1 of 6 possible choices as there are 6 red balls, out of 15 total choices. so there are 6 ways out of 15 to get a red ball.

Since the ball is replaced, when choosing the next ball, you still have 15 total choices, of which you must choose 1. So the probabilities for choosing a green and blue ball are:

and

, respectively.

Since each choice is independent, the probability of all three happening is just the product of the probabilities, that is:

For "without replacement": after picking each ball, the total remaining choices decrease by 1. So after picking the red, there are 14 balls from which to choose the green, then 13 from which to choose the blue. hence this probability is: