The doughnut shop has 5 kinds of doughnuts: a, b, c,d and e. There are unlimited supply of each kind. In how many ways can you order a dozen doughnuts? Well, my first instict is to simply 5^12. But then I realize aaaab is the same as baaaa... hence the order doesnt matter. I'm trying to do it by cases: Case 1: one daughnuts only. (5 C 1) x 1 = 5 ways Case 2: two daughnuts only. (5 C 2) x 13 = 130 But this include one daughnuts, so its 130-5 = 125. The problem starts here. I dont know why the arrangements for 2 daughnuts is 13. I got it by simply listing out all the cases. Then I tried using simpler problems, like let's say you want to buy 5 daughnuts out of 3 different daughnuts. I can find out the number of ways for this one, but I see no relation if let's say you want to buy 6 daughnuts. I sat down for 2 hours and still couldnt figure it out :( My last bet is 13^4... but that's a wild guess. Help is appreciated.