How to find all combinations of coins when given some dollar value
I found a piece of code that I was writing for interview prep few months ago.
According to the comment I had, it was trying to solve this problem:
Given some dollar value in cents (e.g. 200 = 2 dollars, 1000 = 10 dollars), find all the combinations of coins that make up the dollar value. There are only pennies (1¢), nickels (5¢), dimes (10¢), and quarters (25¢) allowed.
For example, if 100 was given, the answer should be:
4 quarter(s) 0 dime(s) 0 nickel(s) 0 pennies
3 quarter(s) 1 dime(s) 0 nickel(s) 15 pennies
etc.
I believe that this can be solved in both iterative and recursive ways. My recursive solution is quite buggy, and I was wondering how other people would solve this problem. The difficult part of this problem was making it as efficient as possible.
Answer
I looked into this once a long time ago, and you can read my little write-up on it. Here’s the Mathematica source.
By using generating functions, you can get a closed-form constant-time solution to the problem. Graham, Knuth, and Patashnik’s Concrete Mathematics is the book for this, and contains a fairly extensive discussion of the problem. Essentially you define a polynomial where the nth coefficient is the number of ways of making change for n dollars.
Pages 4-5 of the writeup show how you can use Mathematica (or any other convenient computer algebra system) to compute the answer for 10^10^6 dollars in a couple seconds in three lines of code.
(And this was long enough ago that that’s a couple of seconds on a 75Mhz Pentium...)