Since 59 ends with 9, we must use either 4 or 9 1c coins.  

The only way we can use 9 1c coins, is 1 50c coin and 9 1c coins.  
And that is the only way we can use a 50c coin.

1.  So that's 1 way, 1 50c coin and 9 1c coins.

For any other ways we must use no 50c coins and 4 1c coins, so we must 
mave 55c with 6 coins.

We can't use 2 20c coins because that would mean having 15c with
4 coins, using 10c and 5c coins, which cannot be done.

If we use 1 20c coin, we must have 35c in 5 coins, which could only 
be had with 2 10c coins and 3 5c coins.

2.  So that's a second way, 1 20c coin, 2 10c coins, 1 5c coin, and,
of course, 4 1c coins.

If we don't use any 50c or 20c coins, we must have 55C in 6 5c and 10c
coins.  Since 55c ends in a 5, we must have 1 5c coin, so we put that 
aside and get the remaining 50c with 5 coins, which we can do by using 
5 10c coins.  

3.  So that's the third way, 1 5c coin and 5 10c coins, and, of course,
4 1c coins. 

Answer: 3 ways.

Edwin