Start with the largest denomination and work your way down, starting with the most of that denomination and work down to none of that denomination.

1) With 1 10p we have 0p left, so one combination with a single 10p. (*).

2) With 2*5p we have 0p left, so one combination with 2*5p (*).

3) With 1*5p we have 5p left, so find number of ways of making 5p with denominations less than 5p:

3A) With 2*2p we have 1p left, so find number of ways of making 1p with denominations less than 2p.

3Aa) With 1p we have 0p left, so one combination with 1*5p, 2*2p, 1*1p (*)

3B) With 1*2p we have 3p left, so find number of ways of making 3p with denominations less than 2p.

3Ba) With 3*1p we have 0p left, so one combination with 1*5p, 1*2p, 3*1p (*)

3Bb) With 2*1p, we have 1p left, clearly no way to make this up as no denominations smaller than 1p.

3Bc) With 1*1p, we have 2p left, clearly no way to make this up as no denominations smaller than 1p.

3C) With no 2ps, we have 5p left, so find number of ways of making 5p with denominations less than 2p.

3Ca) With 5*1p we have 0p left, so one combination with 1*5p, 5*1p (*)

3Cb) Clearly no solutions with less than 5*1p (see 3Bb and 3Bc).

4) With no 5ps we have 10p left, so find number of ways of making 10p with denominations less than 5p.

4A) With 5*2p we have 0p left, so one combination with 5*2p (*)

4B) With 4*2p we have 2p left, so one combination with 4*2p, 2*1p (*)

4C) With 3*2p we have 4p left, so one combination with 3*2p, 4*1p (*)

4D) With 2*2p we have 6p left, so one combination with 2*2p, 6*1p (*)

4E) With 1*2p we have 8p left, so one combination with 1*2p, 8*1p (*)

4F) With no 2ps we have 10p left, so one combination with 10*1p (*)