Question 816182: I want to know how many ways I can make $2 using dollars, half dollars, and quarters. I got this answer:
1. 1 + 1
2. 8 q
3. 2 half + 2 half
4. 1 + 2 half
5. 1 + 4 quarters
6. 1 + 1 half + 2 q
7. 1 dollar + 4 q
8. 2 half + 4 q
9. 3 half + 2 q
I drove myself crazy trying to figure this out, and I want to know if there's a better way to solve this with a chart. What I did was think about, and draw, coins on paper then look relied on just that to figure out how many combinations there were. I think I'm wrong, especially when I used the same method to figure out how many ways I can make $3 using quarters, half dollars, and dollars, and only got 10 different ways (I think). Any help would be sooooooooooooooooo appreciated. Thank you! Jay
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! Here's an approach which may help.
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.)
Let's look at a rectangular array of q's, where each q represents a quarter:
q q q q
q q q q
8q
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).
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
Now repeat the process except if there are a pair of h's replace them with a d:
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
For other amounts of money, just add more columns of q's.
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