Obviously there are quite a few ways to do this including El Bruto (brute force).

In a similar problem, optimization problem, the aim is to find a combination of weights to reach a certain total. With the difference that you have integers and any combination is acceptable. I have suggested using genetic algorithms and there is code there to do that.

Another way, possibly inefficient, is to use a tree.

#!/usr/bin/env perl

# depth-first search on a tree of combinations
# aim is to use @input numbers to make the target sum.
# by bliako
# for https://perlmonks.org/?node_id=11102502
# 08/07/2019

use strict;
use warnings;

# from https://github.com/hadjiprocopis/Tree-Nary-Tiny
# or any other nary-tree implementation, e.g.
# https://metacpan.org/pod/Tree::Simple
use Tree::Nary::Tiny;

#for deep recursion try this:
#my @input = map { 2+int(rand(20)) } 0..100;
#my $target = 15783;

my @input = qw/ 2 5 7 /;
my $target = 15;

my $T = Tree::Nary::Tiny->new(
    undef, # parent
    "root",
    undef, # data
    sub { return $_[0] }
);
my @solutions;
find($T, \@input, $target, \@solutions);

print "$0 : here are all the solutions:\n";
my $i = 1;
foreach (@solutions){
    my $sum = 0;
    $sum += $_ foreach (@$_);
    if( $sum != $target ){ die "wrong solution, this should not be hap
+pening." }
    print $i . ")". join(",", @$_)."\n";
    $i++;
}

sub    find {
    my $n = $_[0];
    my $input = $_[1];
    my $target = $_[2];
    my $solutions = $_[3];
    my $v = $n->value();

    if( defined $v && $v->{'sum'} == $target ){
        my @asol = ();
        while( defined $n->parent() ){
            push @asol, $n->value()->{'number'};
            $n = $n->parent();
        }
        push @$solutions, \@asol;
        print "found solution: ".join(",",@asol)."\n";
        return
    }

    my $sum = defined($v) ? $v->{'sum'} : 0;
    foreach(@$input){
                # added this to make sure that we have combinations ra
+ther than permutations:
                # see haukex's comment below
                next if( defined($v) && $_ < $v->{'number'} );
        if( $sum + $_ <= $target ){
            my $nn = Tree::Nary::Tiny->new(
                $n, # parent
                $_, # an id, nothing important
                {
                    'sum' => $sum + $_,
                    'number' => $_
                }, # the data to hold
            );
            find($nn, $input, $target, $solutions);
        }
    }
}
[download]

tr.pl : here are all the solutions:
1)5,2,2,2,2,2
2)2,5,2,2,2,2
3)7,2,2,2,2
4)2,2,5,2,2,2
5)2,7,2,2,2
6)2,2,2,5,2,2
7)2,2,7,2,2
8)2,2,2,2,5,2
9)2,2,2,7,2
10)2,2,2,2,2,5
11)5,5,5
12)2,2,2,2,7
[download]

EDIT after soonix's comment below: I have added the next if ... statement to my code above like so (so above code now is corrected to give combinations rather than permutations (=order matters)):

        foreach(@$input){
                # added this to make sure that we have combinations ra
+ther than permutations:
                # see haukex's comment below
                next if( defined($v) && $_ < $v->{'number'} );
                ...
[download]

bw, bliako

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