Combining Coins to Make £6

Using only 2 pence, 10 pence and 50 pence coins, in exactly how many ways can 100 coins be made to total £6?
We can write the problem as the pair of simultaneous equations
\[2x+10y+50z=600 \rightarrow x+5y+25z=300\]

\[x+y+z=100\]

where  
\[x, \; y, \; z\]
  are the number of 2 pence, 10 pence and 50 pence coins respectively and only positive integer solutions are required.
Let  
\[z=k\]
  then we can write the above equations as
\[x+5y=300-25k\]
  (1)
\[x+y=100-k\]
  (2)
(2)-(1) gives  
\[4y=200-24k \rightarrow y=50-6k\]
  then from (2)  
\[x=100-k=100-(k)=50+5k\]
.
\[z=k, \; y \ge 0 \rightarrow 0 \le k \le 8 \]
.
Possible values of  
\[x, \; y, \; z\]
  are given in the table.
 
\[k\]
 		  
\[x\]
 		  
\[y\]
 		  
\[z\]
 
0 50 50 0
1 55 44 1
2 60 38 2
3 65 32 3
4 70 26 4
5 75 20 5
6 80 14 6
7 85 8 7
8 90 2 8

You have no rights to post comments