Can someone explain the maths involved in being able to solve this to me please:

Every packet of Captain Crunch Chips comes with a randomly inserted letter: C, H, I, or P. If you can collect one of each letter, you will win a free packet of chips. Each letter is equally likely, what is the expected number of chip packets you need to open to win a free packet?

I can find the probability of getting it in four goes, and five etc with work (probably do a probability tree). And have used a quick simulation to get a result of 7.4, but am sure that there must be a systematic way to solve it. Would also like to then be able to generalize it to a problem that would have more than 4 possible picks etc.

Thanks