We must use at least 1 quarter because four nickels and three dimes
is only 50 cents.

If we use 1 quarter, then we must make the remaining 45c
with nickels and dimes.  We must use all 3 dimes, because
2 dimes and 4 nickels is only 40c. So 1 quarter and 3 dimes
is 55c and we can make the remaining 15c with 3 ninckels.

That's 1 way, 

1 quarter, 3 dimes and 3 nickels. 

If we use 2 quarters, then we must make the remaining 20c
with nickels and dimes.  We can use either

2 quarters, 2 dimes and 0 nickels
2 quarters, 1 dime and 2 nickels
2 quarters, 0 dimes and 4 nickels

That 3 more ways.

Total: 4 ways

Edwin