> I wonder if my way of solving this problem is correct

Nope, sorry :-). First, you missed a bunch of possibilities (e.g., you said that to get in after 2 days you need to choose 2,0, but you could choose 1,1,0 instead; you said that to get in after 3 days you need to choose 1,1,1,0, but you could also choose 1,2,0 or 2,1,0). Second, at best what you've got is an estimate. Third, the question wasn't "what's the most likely time it takes", which I think is what you've tried to work out, but "how long on average". (The most likely time is actually 0 days.)