The question today comes from 1997 Paper I. The original question is shown below.
(i) Show that you can make up 10 pence in eleven ways using 10p, 5p, 2p and 1p coins.
(ii) In how many ways can you can make up 20 pence using 20p, 10p, 5p, 2p and 1p coins?
This kind of question—often shown in primary school mathematics olympiads in China—has a bunch of neat and elegant ways of approaching; nonetheless, it turns out that the bluntest instrument is the quickest—listing all the possibilities, methodically and systematically.
(i) One can make up 10p as follows:
using only 10p: 10p; (1 way)
using no 10p but 5p: 5p+5p; (1 way)
using only 5p: 5p+2p+2p+1p, 5p+2p+1p+1p+1p, 5p+1p+1p+1p+1p+1p; (3 ways)
using neither 10p nor 5p: 2p+2p+2p+2p+2p, or to substitute 1/2/3/4/5 2p's with 1p+1p. (6 ways)
There are therefore 1+1+3+6=11 ways.
(ii) Considering 20p = 10p + 10p, one can make up 20p as follows:
using only 20p: 20p; (1 way)
using one 10p: 10p+any arrangement in (i); (11 ways)
using 4 5p's: 5p+5p+5p+5p; (1 way)
using 3 5p's: 5p+5p+5p+2p+2p+1p, 5p+5p+5p+2p+1p+1p+1p, 5p+5p+5p+1p+1p+1p+1p+1p; (3 ways)
The reader is advised to consider the remaining three cases.
using 2 5p's: 5p+5p+...; (6 ways)
using 1 5p's: 5p+...; (8 ways)
using no 5p's. (11 ways)
There are therefore 1+11+1+3+6+8+11=41 ways.