Case 1:  If we don't use any quarters, then we must make the whole
48 cents with dimes, nickels and pennies.

A. If we use 0 dimes we can use 0 up to 9 nickels and the rest pennies.
   That's 10 ways.

B. If we use 1 dime we can use 0 up to 7 nickels and the rest pennies.
   That's 8 ways.

C. If we use 2 dimes we can use 0 up to 5 nickels and the rest pennies.
   That's 6 ways.
 
D. If we use 3 dimes we can use 0 up to 3 nickels and the rest pennies.
   That's 4 ways.

D. If we use 4 dimes we can use 0 or 1 nickel and the rest pennies.
   That's 2 ways.

So with no quarters there are 10+8+6+4+2 or 30 ways 

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Case 2:  If we use 1 quarter, we have to make the remaining 23 cents with
dimes, nickels, and pennies 

A. If we use 0 dimes we can use 0 up to 4 nickels and the rest pennies.
   That's 5 ways.

B. If we use 1 dime we can use 0 up to 2 nickels and the rest pennies.
   That's 3 ways.

C. If we use 2 dimes we can only use 3 pennies.
   That's 1 way.

So with 1 quarter there are 5+3+1 or 9 ways to make the remaining 23 cents.

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              30 ways using 0 quarters
               9 ways using 1 quarter
              -----------------------
      total:  39 ways to have 48 cents in coin. 

Edwin