I would just add the 2 distributions so you would let Z = X + Y so then Z~Po(5)
Then use your method to find the P(Z=2), it should give the answer the book gives.
Also, your method could be right (I'm not sure) as there are 2 ways in which X and Y can happen, X first then Y or Y first then X. You only did it one of those ways which would explain your answer being half the answer in the book.