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Change for a Dollar

Date: 03/07/99 at 13:32:09
From: jay
Subject: Ways to make a change for a dollar?

How many different ways are there to make a change for a dollar?


Date: 03/26/99 at 00:50:08
From: Doctor Schwa
Subject: Re: Ways to make a change for a dollar?

This problem is discussed in _How To Solve It_, by George Polya, which 
is a book I'd certainly recommend for anyone.

A computer program that solves this problem is in our Dr. Math archives
(I found it by searching for "change dollar ways") at


Some more hints can be found at


Another method:

Make a list of the number of ways to make change using only pennies;
there's only one way, namely all pennies.

   5  10  15  20  25  30  35  40  45  50
P  1   1   1   1   1   1   1   1   1   1

Then try making a list of the number of ways to do it with nickels and

With 5 cents, there are 2 ways: a nickel or 5 pennies.

With 10 cents, either you use NO nickels (1 way) or you use a nickel
and then make the remaining 5 cents (2 ways).  Total, 3 ways.

Keep going like that, and you'll get a row like

NP  2   3   4 ....

Then you can include dimes, and make a DNP row like
DNP 2   4   6  ...

How did I figure out those numbers? Well, for 5 cents, dimes are 
useless, so it's the same as the NP row. For 10 cents, a dime gives you 
one extra way compared to the NP row. For 15 cents, you can use NO 
dimes (4 ways; that comes from the NP row) or you can use a dime and 
then make the remaining 5 cents somehow (2 ways, from the DNP row), for
a total of 6 ways.

Continuing on like that, you can make a complete table of the number of 
ways to make change for n cents using just pennies, or allowing 
nickels, or dimes, or quarters....

Have fun!

- Doctor Schwa, The Math Forum
Associated Topics:
Elementary Puzzles
High School Permutations and Combinations
High School Puzzles
Middle School Puzzles

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