SOLUTION: How many combinations can be made into 41 cents only using pennies, quarters, dimes, and nickles

31 ways.  Here's why:

You can either use no quarter, or 1 quarter.

Case 1:  you don't use a quarter.

1. If you don't use any dimes, then you can use from 0 through 8 nickels 
and the rest pennies.  That's 9 ways.  

2. If you use exactly 1 dime, you can use from 0 through 6 nickels 
and the rest pennies.  That's 7 ways.

3. If you use exactly 2 dimes, you can use from 0 through 4 nickels 
and the rest pennies.  That's 5 ways.

4. If you use exactly 3 dimes, you can use from 0 through 2 nickels 
and the rest pennies.  That's 3 ways

5. If you use exactly 4 dimes, you must use just 1 penny.  
That's 1 way.

For case 1, where there are no quarters. there are 9+7+5+3+1 or
25 ways.

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Case 2:  you use 1 quarter.

Now the problem is to make the remaining 16 cents using dimes, 
nickels and pennies. You can either use no dime or 1 dime.

1. If you don't use a dime, you can use from 0 through 3 nickels 
and the rest pennies.  That's 4 ways.

2. If you use exactly 1 dime, you can use no nickel or 1 nickel. 
and the rest pennies.  That's 2 ways.

For case 2, where there is 1 quarter. there are 4+2 or 6 ways to
make the remaining 16 cents.

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Final answer: 25+6 or 31 ways. 

Edwin