How many ways can you make change: Some easy proofs

TL;DR

This paper presents simplified proofs for counting the number of ways to make change with various coin sets, using recurrences and generating functions, enhancing understanding of a classical combinatorial problem.

Contribution

It introduces simpler proof techniques for the change-making problem, including recurrences for specific coin sets and generating functions for general cases.

Findings

Derived formulas for coin sets {1,x,kx,rx} using recurrences

Used generating functions to obtain formulas for arbitrary coin sets

Provided clearer, more accessible proofs for classical change-making formulas

Abstract

Given a dollar, how many ways are there to make change using pennies, nickels, dimes, and quarters? What if you are given a different amount of money? What if you use different coin denominations? This is a well known problem and formulas are known. We present simpler proofs in several cases. We use recurrences to derive formulas if the coin denominations are {1,x,kx,rx}, and we use a simple proof using generating functions to derive a formula for any coin set.