Question 816182
Here's an approach which may help.<br>
The quarters can only come in pairs. Using just dollars, half-dollars and quarters, there is no way to get $2 exactly with an odd number of quarters. (Think about it.)<br>
Let's look at a rectangular array of q's, where each q represents a quarter:
<pre>
  q   q   q   q
  q   q   q   q
8q
</pre>These 8 quarters add up to $2. Each column of two quarters is equal to a half-dollar. Each row of 4 quarters is equal to a dollar. To find all the combinations: First count the initial array of 8 quarters. Then start replacing columns with h's (for half-dollars).
<pre>
  q   q   q   h      
  q   q   q   
6q + 1h

  q   q   h   h      
  q   q      
4q + 2h

  q   h   h   h      
  q         
2q + 3h

  h   h   h   h               
4h
</pre>Now repeat the process except if there are a pair of h's replace them with a d:
<pre>
  q   q   q   q
  q   q   q   q
No pairs of h's

  q   q   q   h      
  q   q   q   
No pairs of h's

  q   q   h   h      
  q   q      
Replace the pair of h's:
  q   q   d
  q   q    
4q + 1d  

  q   h   h   h      
  q         
Replace a pair of h's:
  q   h   d      
  q        
2q + 1h + 1d 

  h   h   h   h               
Replace one pair of h's:
  h   h   d
2h + 1d
Replace the remaining pair of h's:
  d   d
2d
</pre>For other amounts of money, just add more columns of q's.