20 coins, all nickels, is a dollar.

Since there is exactly 1 nickel, she has to have 
the remaining 95 cents with 19 coins.

Obviously the number of pennies has to be 0, 5, 10, or 15

She can't have 0 pennies because even 19 dimes 
is way more than 95 cents, let alone having any
higher coins.

If she had only 5 pennies, then those and the nickel 
make 10 cents with 6 coins, and the remaining 90 cents 
must be made with 14 coins. But even 14 dimes is way
more than 90 cents, let alone having any higher 
coins.

If she had only 10 pennies, then those and the nickel 
make 15 cents with 11 coins, and the remaining 85 cents 
must be made with 9 coins. But even 9 dimes is more 
than 85 cents, let alone any higher coins.

So she must have exactly 15 pennies, then those and the 
nickel make 20 cents with 16 coins, and the remaining 80 
cents must be made with 4 coins.  

If she had exactly 1 quarter, she'd have to have 55
cents with 3 coins.  There is no way she could have that 
without having another nickel.

If she had exactly 2 quarters, she'd have to make 30
cents with 2 coins.  There is no way she could have that 
without having another nickel. 

If she had exactly 3 quarters, she'd have to have 5
cents with 1 coin.  But she can't have another nickel.
So she can't have any quarters.

So the only way she can have that is with 1 half dollar
and 3 dimes.  So that's the answer:

She has 1 half dollar, 3 dimes, 1 nickel and 15 pennies.

[Half dollars aren't in circulation anymore, but Mary 
in the problem has one!  I have a few myself.  I'll 
bet you've got one stashed in a drawer someplace in 
your house!  :)  ] 

Edwin