**change_polynomial**,
a Python code which
uses a polynomial multiplication algorithm to count the ways of making
various sums using a given number of coins.

In this version of the change making problem, the order of the coins is significant. In other words, if we are using exactly 3 coins to make 16 cents, then we count 6 separate solutions: (1,5,10), (1,10,5), (5,1,10), (5,10,1), (10,1,5) and (10,5,1). Similarly, we count three essentialy equivalent solutions to forming the sum of 7 cents.

If the user wishes to determine the number of ways of forming sums using anywhere from 0 up to COIN_NUM coins, inclusive, then it is only necessary to include a coin of value 0 in the denomination list.

The computer code and data files described and made available on this web page are distributed under the MIT license

**change_polynomial** is available in
a MATLAB version and
an Octave version and
a Python version.

change_diophantine, a Python code which sets up a Diophantine equation to solve the change making problem, which counts the number of ways a given sum can be formed using coins of various denominations.

change_dynamic, a Python code which uses dynamic programming to solve the change making problem, which counts the number of ways a given sum can be formed using coins of various denominations.

change_greedy, a Python code which uses the greedy method to seek a solution to the change making problem, which tries to match a given amount by selecting coins of various denominations.

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- change_polynomial.py, the source code.
- change_polynomial.sh, runs all the tests.
- change_polynomial.txt, the output file.