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 $1.05 (105 cents), if at least one of each type of coin must be included?
Choose the correct choice from the following options
A) 49 B) 69 C) 73 D) 91 E) 99
I. Case I = Four Quarters = this is impossible, because if we already have a dollar in quarters, then we cannot have any dimes. We can't have all four coins represented if we have four quarters.   II. Case II = Three Quarters (75 cents)   This allows for two or one dime. Three dimes would bring us up to $1.05, and we wouldn't have room for any pennies or nickels.   Subcase 1 = 3 Q, 2 D (95 cents)   1) one nickel (N) and 5 pennies (P) (2 nickels would not allow for any pennies)   Subcase 2 = 3 Q, 1 D (85 cents)   2) 3 N and 5 P   3) 2 N and 10 P   4) 1 N and 15 P   Case II allows for a total of four ways.   III. Case III = Two Quarters (50 cents)   This allows for 1-4 dimes.   Subcase 1 = 2 Q, 4 D (90 cents)   (2 ways) we could have 1-2 nickels   Subcase 2 = 2 Q, 3 D (80 cents)   (4 ways) we could have 1-4 nickels   Subcase 3 = 2 Q, 2 D (70 cents)   (6 ways) we could have 1-6 nickels   Subcase 4 = 2 Q, 1 D (60 cents)   (8 ways) we could have 1-8 nickels   Case III allows for a total of 20 ways   IV. Case IV = One Quarter (25 cents)   This allows for 1-7 dimes   Subcase 1 = 1 Q, 7 D (95 cents)   (1 way) we could have 1 nickel   Subcase 2 = 1 Q, 6 D (85 cents)   (3 ways) we could have 1-3 nickels   Subcase 3 = 1 Q, 5 D (75 cents)   (5 ways) we could have 1-5 nickels   Subcase 4 = 1 Q, 4 D (65 cents)   (7 ways) we could have 1-7 nickels   Subcase 5 = 1 Q, 3 (55 cents)   (9 ways) we could have 1-9 nickels   Subcase 6 = 1 Q, 2 D (45 cents)   (11 ways) we could have 1-11 nickels   Subcase 7 = 1 Q, 1 D (35 cents)   (13 ways) we could have 1-13 nickels   Case IV allows for a total of 49 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 = 4 + 20 + 49= 
<gadget id="calculator">4 + 20 + 49</gadget>
<output>73</output>
73 ways.   OA=(73)
<result>C</result>
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 $1.05 (105 cents), if at least one of each type of coin must be included?