How many ways are there to write 14 as a sum of 5 positive integers, not necessarily distinct? Order does not matter, like 1+1+1+1+10 is the same as 10+1+1+1+1.
This is a "partition" problem of 14 into 5 parts. There are 23 such partitions as follows:
10 + 1 + 1 + 1 + 1 = 14
9 + 2 + 1 + 1 + 1 = 14
8 + 3 + 1 + 1 + 1 = 14
8 + 2 + 2 + 1 + 1 = 14
7 + 4 + 1 + 1 + 1 = 14
7 + 3 + 2 + 1 + 1 = 14
7 + 2 + 2 + 2 + 1 = 14
6 + 5 + 1 + 1 + 1 = 14
6 + 4 + 2 + 1 + 1 = 14
6 + 3 + 3 + 1 + 1 = 14
6 + 3 + 2 + 2 + 1 = 14
6 + 2 + 2 + 2 + 2 = 14
5 + 5 + 2 + 1 + 1 = 14
5 + 4 + 3 + 1 + 1 = 14
5 + 4 + 2 + 2 + 1 = 14
5 + 3 + 3 + 2 + 1 = 14
5 + 3 + 2 + 2 + 2 = 14
4 + 4 + 4 + 1 + 1 = 14
4 + 4 + 3 + 2 + 1 = 14
4 + 4 + 2 + 2 + 2 = 14
4 + 3 + 3 + 3 + 1 = 14
4 + 3 + 3 + 2 + 2 = 14
3 + 3 + 3 + 3 + 2 = 14
