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You can put this solution on YOUR website!
\n" ); document.write( "56.
\n" ); document.write( "But my showing you the whole process for finding the answer doesn't do you any good if you want to learn anything from the problem.
\n" ); document.write( "So I'll show you how you can go about finding the answer and let you see if you come up with the right answer doing the work yourself.
\n" ); document.write( "The number of $1 bills must be a multiple of 5. So break the problem into cases for the different possible numbers of $1 bills:
\r\n" ); document.write( "if there are 0 $1 bills, the amount remaining is $50; the number of ways to make $50 using $20, $10, and $5 bills is ______\r\n" ); document.write( "if there are 5 $1 bills, the amount remaining is $45; the number of ways to make $45 using $20, $10, and $5 bills is ______\r\n" ); document.write( "if there are 10 $1 bills, the amount remaining is $40; the number of ways to make $40 using $20, $10, and $5 bills is ______\r\n" ); document.write( "...\r\n" ); document.write( "if there are 40 $1 bills, the amount remaining is $10; the number of ways to make $10 using $20, $10, and $5 bills is ______\r\n" ); document.write( "if there are 45 $1 bills, the amount remaining is $5; the number of ways to make $5 using $20, $10, and $5 bills is ______\r\n" ); document.write( "if there are 50 $1 bills, the amount remaining is $0; the number of ways to make $0 using $20, $10, and $5 bills is ______
\n" ); document.write( "So you have 11 cases to work separately. However, if you are careful and organized with your work, you can see a pattern that helps you complete the problem without working out all 11 cases completely.
\n" ); document.write( "Here are the workings for a couple of the cases.
\n" ); document.write( "Case 1: Suppose there are 10 $1 bills; that leaves $40 to be made with $20, $10, and $5 bills.
\n" ); document.write( "If there are 2 $20 bills, that is the whole remaining $40; there is only 1 way to complete this case using 2 $20 bills.
\n" ); document.write( "If there is 1 $20 bill, that leaves $20. There are 3 choices for the number of $10 bills to use to make that remaining $20 -- 0, 1, or 2. Whatever isn't made up with the $10 bills will be made with the $5 bills. So there are 3 ways to complete this case using 1 $20 bill.
\n" ); document.write( "If there are no $20 bills, that leaves $40; that gives 5 choices for the number of $10 bills to use -- 0, 1, 2, 3, or 4; and again the $5 bills will make up whatever the $10 bills don't. So there are 5 ways to complete this case using no $20 bills.
\n" ); document.write( "So for the case where there are 10 $1 bills, the number of ways to make change for $50 is 1+3+5 = 9.
\n" ); document.write( "Case 2: Suppose there are 35 $1 bills, leaving $15 to be made using the $20, $10, and $5 bills.
\n" ); document.write( "Obviously we can't have any $20 bills; so we need to make the remaining $15 with only $10 and $5 bills. As before, we have 2 choices for the number of $10 bills to use -- 0 or 1 -- and the $5 bills will make up what is left.
\n" ); document.write( "So for the case where there are 35 $1 bills, there are only 2 ways to make change for the $50 bill.
\n" ); document.write( "Do the similar analyses for the other cases to find the answer.\r
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