Problem source

Show that you can make up 10 pence in eleven ways using 10p, 5p, 2p and 1p coins.
In how many ways can you make up 20 pence using 20p, 10p, 5p, 2p and 1p coins?
You are reminded that no credit will be given for unexplained answers.

Solution source

Writing out the possibilities in order of the largest coin used (and then second largest and so-on):

\begin{align*}
&& 10 &= 10 \\
&&&= 5 + 5 \\
&&&= 5 + 2 + 2 + 1 \\
&&&= 5 + 2 + 1 + 1 + 1 \\
&&&= 5 + 1 + 1 + 1 + 1 + 1\\
&&&= 2 + 2 + 2 + 2 + 2 = 5 \cdot 2\\
&&&= 4 \cdot 2 + 2 \cdot 1 \\
&&&= 3 \cdot 2 + 4 \cdot 1\\
&&&= 2 \cdot 2 + 6\cdot 1\\
&&&= 1 \cdot 2 + 8\cdot 1 \\
&&&= 10 \cdot 1
\end{align*}

For 20p, we have

\begin{align*}
&& 20 &= 20 \\
&&&= 10 + \text{all 11 ways} \\
&&&= 4\cdot 5 \\
&&&= 3\cdot 5 +\text{3 ways} \\
&&&= 2\cdot5 + \text{6 ways} \\
&&&= 1\cdot 5 + \text{8 ways} \\
&&&= k\cdot 2 + (20-2k)\cdot 1 \quad \text{11 ways} 
\end{align*}

ie 41 ways