In how many different ways can change be provided for a one dollar bill using pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents), if at least one of each type of coin must be included?	Pick one
A) 39 B) 42 C) 60 D) 65 E) 66
I. Case I = Four Quarters = this is impossible, because if we already have a dollar in quarters, then we cannot have any other coins. We can't have all four coins represented if we have four quarters.   Case I does not allow any ways.   II. Case II = Three Quarters (75 cents)   This allows for one dime. Two dimes would bring us up to $0.95, and we wouldn't have room for both pennies and nickels.   Subcase 1 = 3 Q, 1 D (85 cents)   1) one nickel (N) and 10 pennies (P)   2) 2 N and 5 P   Case II allows for a total of two ways.   III. Case III = Two Quarters (50 cents)   This allows for 1-4 dimes.   Subcase 1 = 2 Q, 4 D (90 cents)   (1 way) we could have 1 nickel   Subcase 2 = 2 Q, 3 D (80 cents)   (3 ways) we could have 1-3 nickels   Subcase 3 = 2 Q, 2 D (70 cents)   (5 ways) we could have 1-5 nickels   Subcase 4 = 2 Q, 1 D (60 cents)   (7 ways) we could have 1-7 nickels   Case III allows for a total of 16 ways   IV. Case IV = One Quarter (25 cents)   This allows for 1-6 dimes   Subcase 1 = 1 Q, 6 D (85 cents)   (2 ways) we could have 1-2 nickels   Subcase 2 = 1 Q, 5 D (75 cents)   (4 ways) we could have 1-4 nickels   Subcase 3 = 1 Q, 4 D (65 cents)   (6 ways) we could have 1-6 nickels   Subcase 4 = 1 Q, 3 D (55 cents)   (8 ways) we could have 1-8 nickels   Subcase 5 = 1 Q, 2 D (45 cents)   (10 ways) we could have 1-10 nickels   Subcase 6 = 1 Q, 1 D (35 cents)   (12 ways) we could have 1-12 nickels   Case IV allows for a total of 42 ways.   There's no other case, because we have to have at least one quarter and one dime. The total over the cases equals   Total = 0 + 2 + 16 + 42= 
<gadget id="calculator">0 + 2 + 16 + 42</gadget>
<output>60</output>
60 ways.   OA=(60)
<result>C</result>
In how many different ways can change be provided for a one dollar bill using pennies (1 cent), nickels (5 cents), dimes (10 cents), and quarters (25 cents), if at least one of each type of coin must be included?