There are 292 ways to break a dollar using half-dollars, quarters, dimes, nickels, and cents. The following snippet lists them all. Use 
% perl make_change.pl 100 50 25 10 5 1

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More generally, if you want to find out how many ways one can break n into chunks of size m1,...,mk, then the following command will print them out:

% perl make_change.pl n m1 ... mk

Update: Spurred by Roy Johnson's reply++ I tightened my version slightly to take care of edge cases in a more natural way, and explicitly list the 0 coefficients. Originally I thought I liked the looks of the "zero-less" output better, but on a second thought I realized that a version of the core function that returns all the coefficients is more generally useful; the output can be easily tweaked by the calling code. The current version lists all the coefficients, one for each coin.

Keyword: partitions.

use strict;
use warnings;

sub make_change {
  my $amount  = shift;
  return $amount ? () : [] unless @_;
  my $coin    = pop;
  my $n_coins = 0;
  my $base    = 0;
  my @ways;
  while ( $base <= $amount ) {
    push @ways, map [ @$_, [ $n_coins, $coin ] ],
                make_change( $amount - $base, @_ );
    $n_coins++;
    $base += $coin;
  }
  return @ways;
}

sub print_way {
  my $way = shift;
  print join ' + ', map "$_->[ 0 ] x $_->[ 1 ]", @$way;
  print $/;
}

print_way( $_ ) for make_change( @ARGV );

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In reply to How many ways to make change? by tlm

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