It seemed a little long, so I just tweaked it a tiny bit :)

#!/usr/bin/perl

# https://perlmonks.org/?node_id=1228495

use strict;
use warnings;

my @results = Solve(869,
  [15, 43, 51, 56, 60, 67, 122, 152, 193, 204, 229, 271, 293, 301] );
print "@$_\n" for @results;

sub Solve
  {
  my $want = shift() or return [];
  $want > 0 or return ();
  my ($first, @rest) = @{ shift() } or return ();
  map([$first, @$_], Solve($want - $first, \@rest)), Solve($want, \@re
+st);
  }
[download]

Outputs:

15 43 51 56 60 122 229 293
15 43 51 56 204 229 271
15 43 51 67 193 229 271
15 43 51 122 152 193 293
15 56 60 67 122 152 193 204
15 56 60 122 152 193 271
15 56 204 293 301
15 67 193 293 301
43 51 56 67 122 229 301
43 51 56 67 152 229 271
43 51 60 193 229 293
43 122 204 229 271
56 60 67 122 271 293
56 67 152 293 301
[download]