Coins Coins Coins

[ShaheerH446]

I came across this and the solution literally adds each possible combination. Is there a better way to solve the question?

In how many different ways can pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents) be combined for a total of $1.10 (110 cents), if at least one of each type of coin must be included?
(A) 36

(B) 51

(C) 70

(D) 87

(E) 101

[Sage Pearce-Higgins]

Just looking at the answers tells you that thinking of all the cases separately is going to be too time-consuming. We need to look for patterns. Before I go into more detail, please can you provide the source for this question?