There are three major cases,
Major case I. Using no half dollars
Major case II. Using exactly 1 half dollar.
Major case III. Using two half dolars.
I. Using 0 half dollars.
There are 4 sub-cases.
Sub-Case A. Using 0 quarters,
Sub-Case B. Using exactly 1 quarter
Sub-Case C. Using exactly 2 quarters
Sub-Case D. Using exactly 3 quarters
Sub-Case E. Using exactly 4 quarters.
Sub-Case A:
If you don't use any quarters you can use from 0 up to 10 dimes, and the
rest, if any, can be had all in nickels. That's 11 possible ways to use
no quarters.
Sub-Case B:
If you use exactly 1 quarter you can use from 0 up to 7 dimes, and the
rest can be had all in nickels. That's 8 possible ways to use
1 quarters.
Sub-Case C:
If you use exactly 2 quarters you can use from 0 up to 5 dimes, and the
rest, if any, can be had all in nickels. That's 6 possible ways to use
2 quarters.
Sub-Case D:
If you use exactly 3 quarters you can use from 0 up to 2 dimes, and the
rest can be had all in nickels. That's 3 possible ways to use
3 quarters.
Sub-Case E:
If you use 4 quarters, that's the whole dollar, so that's 1 more way.
Total from all those sub-cases for Major Case I:
11+8+6+3+1 = 29 ways
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Major case II. Using exactly 1 half dollars. That leaves 50 cents.
There are 3 sub-cases.
Sub-Case A. Using 0 quarters,
Sub-Case B. Using exactly 1 quarter
Sub-Case C. Using exactly 2 quarters
Sub-Case A:
If you don't use any quarters you can use from 0 up to 5 dimes, and the
rest, if any, can be had all in nickels. That's 6 possible ways to use
no quarters.
Sub-Case B:
If you use exactly 1 quarter you can use from 0 up to 2 dimes, and the
rest can be had all in nickels. That's 3 possible ways to use
1 quarters.
Sub-Case C:
If you use exactly 2 quarters you have the whole dollar, so that's
1 more way.
Total from all those sub-cases for Major Case II:
6+3+1 = 10 ways
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Major case III. Using 2 half dollars.
That's the whole dollar. So that's one more case.
So from major case I, II, and III, we have 29+10+1 = 40 ways.
Answer: 40 ways to change a dollar using only half dollars, quarters,
dimes and nickels, no pennies. If you count changing a paper dollar for
a dollar coin, then there are 41 ways.
So the answer is 40 if you don't count 1 silver dollar as a way to change
a dollar, and 41 if you do.
Edwin
