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### Change for a Dollar

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Date: 14 Mar 1995 11:00:30 -0500
From: Anonymous
Subject: Math

How many different ways can you give change for a dollar?
Is there a formula?  Thanks.
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Date: 27 Mar 1995 13:08:36 -0500
From: Dr. Sydney
Subject: Re: Math

Hello!

You ask a good question, and I think you will find it fun to
to suggest how you might find the answer.

One way to approach a problem like this is to deal with
cases.  By that, I mean that you should figure out how many
possible cases there are in a systematic way.  So, let's start
with the biggest change we have.  For \$1, we can give 2 half
dollars.  That is case 1.  Or, we could separate one of those
half dollars, say into 2 quarters (Next you would say 5 dimes,
then you would say 10 nickels, then 50 pennies, then figure
out what all the combinations of pennies, nickels and dimes
would give you 50 cents, and test those).  We could then
separate one of those quarters into 2 dimes and a nickel, or 5
nickels, ..... and so on.

Do you see how you might go about figuring this out?  It would
take a long time and it would be best to write it out in some
kind of organized chart, I think.  Unfortunately , I don't know
of a nice formula for figuring out how many ways there are to
give change for a dollar.

Another way you could approach this problem is to think of it
in terms of the number 5.  Here is what I mean....For the moment
ignore the existence of pennies.  Then, all you have to deal with
are nickels, dimes, quarters, and half-dollars, right?  Well, you
could figure out what are the different combos of 5's that
produce 100, and then figure out for each combo of fives what
change will work (when doing this, take into account the
existence of pennies again!).  For instance, 5*2 + 5*3 + 5*15 = 100,
so for the first 5*2 we could have a dime or 2 nickels or 10
pennies or 5 pennies and a nickel.  Figure out the possiblities
for the 5*3 and the 5*15, and then write down all the possible
combinations of these combinations.  Do this for all combos of
5's you can think of, and you should have a list of all possible
ways you can give change for a dollar (be careful though,
duplication will happen!).

I hope this helps you get an idea of how to approach the problem.
If you are confused by anything, write back and we'll try to clear
it up!

--Sydney, "Dr. Math"
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