Question 1133648: Using only $1,$5,$10 and $20 bills, how many ways can change be made for a $50 bill?
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
56.
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.
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.
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:
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 ______
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 ______
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 ______
...
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 ______
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 ______
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 ______
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.
Here are the workings for a couple of the cases.
Case 1: Suppose there are 10 $1 bills; that leaves $40 to be made with $20, $10, and $5 bills.
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.
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.
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.
So for the case where there are 10 $1 bills, the number of ways to make change for $50 is 1+3+5 = 9.
Case 2: Suppose there are 35 $1 bills, leaving $15 to be made using the $20, $10, and $5 bills.
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.
So for the case where there are 35 $1 bills, there are only 2 ways to make change for the $50 bill.
Do the similar analyses for the other cases to find the answer.
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