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 two of each type of coin must be included?	Choose the most appropriate option:
A) 0 B) 6 C) 8 D) 10 E) 15
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 cannot 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 does not allow for two each of the other coins.   Case II does not allow any ways.   III. Case III = Two Quarters (50 cents)   This allows for 2-3 dimes.   Subcase 1 = 2 Q, 3 D (80 cents)   (2 ways) we could have 2-3 nickels   Subcase 2 = 2 Q, 2 D (70 cents)   (4 ways) we could have 2-5 nickels   Case III allows for a total of 6 ways   There's no other case, because we have to have at least two quarters and two dimes. The total over the cases equals   Total = 0 + 0+ 6= 
<gadget id="calculator">0 + 0+ 6</gadget>
<output>6</output>
6 ways.   OA=(6)
<result>B</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 two of each type of coin must be included?