I have 16 counters. On each turn I either remove 1, 2 or 3 counters until there are no counters left.

I could remove 2 counters eight times, or I could remove five lots of 3 counters then a single counter, or I could remove 3, 2, 2, 1, 3, 1, 1, 1, 1, 1, or any number of different ways.

Well in fact not ‘any’ number of different ways. Your task is to find out exactly how many ways there are. Note: the same combination but occurring in a different order counts as different ways, for instance, there is only one way using 2 eight times, but there are six ways involving five 3s and a 1.

I’ve chosen the number of counters to be 16 because the total number of ways is the square of a prime number. This fact doesn’t particularly help you, except for checking whether or not you are correct.

How many different ways are there?