If we're talking about a monetary system with 1 pence, 2 pence, 5 pence, 10 pences, and 20 pence coins only, then:
The answer appears to be 128 ways. Work shown below.
It's been a while since I've done one of these, so anyone can feel free to correct me if I made a mistake.
If you want to include half-pence, the number of ways will be larger.
If we look at just the 1 pence and 2 pences coins then there are 17 ways using from 0 to 16 of the 2 pence coins, with the rest being 1 pence coins.
If n is even, the # of ways to make a total of n pence using just 1 pence or 2 pence coins is (n+2)/2.
If n is odd, the # of ways would be (n+1)/2.
What are the ways we can use 5 pence, 10 pence, and 20 pence coins to make 32 pence? (the balance will be made with 1 pence and 2 pence coins)
Table: # of 20 pence, # of 10 pence, # of 5 pence, total, amount left to make 32
1x20, 1x10, 0x5, 30, 2
1x20, 0x10, 2x5, 30, 2
1x20, 0x10, 1x5, 25, 7
1x20, 0x10, 0x5, 20, 12
0x20, 3x10, 0x5, 30, 2
0x20, 2x10, 2x5, 30, 2
0x20, 2x10, 1x5, 25, 7
0x20, 2x10, 0x5, 20, 12
0x20, 1x10, 4x5, 30, 2
0x20, 1x10, 3x5, 25, 7
0x20, 1x10, 2x5, 20, 12
0x20, 1x10, 1x5, 15, 17
0x20, 1x10, 0x5, 10, 22
0x20, 0x10, 6x5, 30, 2
0x20, 0x10, 5x5, 25, 7
0x20, 0x10, 4x5, 20, 12
0x20, 0x10, 3x5, 15, 17
0x20, 0x10, 2x5, 10, 22
0x20, 0x10, 1x5, 5, 27
0x20, 0x10, 0x5, 0, 32
So that's 6 ways with 2 pence left, 4 ways with 7 pence left, 4 ways with 12 pence left, 2 ways with 17 pence left, 2 ways with 22 pence left, 1 way with 27 pence left, and 1 way with 32 pence left
For each of those amounts left (2, 7, 12, 17, 22, 27, 32) we can use (n+2)/2 and (n+1)/2 to figure out how many ways to make the remaining amounts with 1 pence and 2 pence coins
6 * (4/2) = 12
4 * (8/2) = 16
4 * (14/2) = 28
2 * (18/2) = 18
2 * (24/2) = 24
1 * (28/2) = 14
1 * (32/2) = 16
The total of 12 + 16 + 28 + 18 + 24 + 14 + 16 is 128.