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