It is Tim's 4 th birthday and he has a cake with 4 candles on it. Tim makes a wish and tries to blow out the candles. Every time he blows, Tim blows out some number of candles between 1 and the number that remain, including both 1 and that number. How many ways are there for Tim to blow out the candles on the cake?
+6251 \(\text{It sounds like your asking for the number of ways you can partition the integer 4}\\ \text{using the digits 1, 2, 3, and 4}\\ (1,1,1,1)\\ (2,1,1) \text{ and it's other 2 permutations}\\ (2,2)\\ (1,3) \text{ and it's other permutation}\\ (4)\\ \text{for a total of $1+ 3+1+2+1 = 8$}\)
.
