That's what arrays and loops are for!

    my @c = split /,/, $hand;
    my $total = 0;

    # 2 cards
    for my $i1 (0..4) {
       for my $i2 ($i1+1..4) {
          $total += 2 if $c[$i1] + $c[$i2] == 15;
       }
    }

    # 3 cards
    for my $i1 (0..4) {
       for my $i2 ($i1+1..4) {
          for my $i3 ($i2+1..4) {
             $total += 2 if $c[$i1] + $c[$i2] + $c[$i3] == 15;
          }
       }
    }

    ...
[download]

But we can do better.

    my @c = split /,/, $hand;
    my $total = 0;

    for my $i1 (0..4) {
        for my $i2 ($i1+1..4) {
            $total += 2 if $c[$i1] + $c[$i2] == 15;
            for my $i3 ($i2+1..4) {
                $total += 2 if $c[$i1] + $c[$i2] + $c[$i3] == 15;
                for my $i4 ($i3+1..4) {
                    $total += 2 if $c[$i1] + $c[$i2] + $c[$i3] + $c[$i
+4] == 15;
                    for my $i5 ($i4+1..4) {
                        $total += 2 if $c[$i1] + $c[$i2] + $c[$i3] + $
+c[$i4] + $c[$i5] == 15;
                    }
                }
            }
        }
    }
[download]

Let's go further:

    my @c = split /,/, $hand;
    my $total = 0;

    our $sum;

    for my $i1 (0..4) {
        local $sum = $i1;
        for my $i2 ($i1+1..4) {
            local $sum = $sum + $c[$i2]];
            $total += 2 if $sum == 15;
            for my $i3 ($i2+1..4) {
                local $sum = $sum + $c[$i3];
                $total += 2 if $sum == 15;
                for my $i4 ($i3+1..4) {
                    local $sum = $sum + $c[$i4];
                    $total += 2 if $sum == 15;
                    for my $i5 ($i4+1..4) {
                        local $sum = $sum + $c[$i5];
                        $total += 2 if $sum == 15;
                    }
                }
            }
        }
    }
[download]

Let's add optimizations:

    my @c = split /,/, $hand;
    my $total = 0;

    our $sum;

    for my $i1 (0..4) {
        local $sum = $i1;
        for my $i2 ($i1+1..4) {
            local $sum = $sum + $c[$i2]];
            $total += 2 if $sum == 15;
            next if $sum >= 15;
            for my $i3 ($i2+1..4) {
                local $sum = $sum + $c[$i3];
                $total += 2 if $sum == 15;
                next if $sum >= 15;
                for my $i4 ($i3+1..4) {
                    local $sum = $sum + $c[$i4];
                    $total += 2 if $sum == 15;
                    next if $sum >= 15;
                    for my $i5 ($i4+1..4) {
                        local $sum = $sum + $c[$i5];
                        $total += 2 if $sum == 15;
                    }
                }
            }
        }
    }
[download]

While I eliminated a lot of redundancy both visually and in the number of checks, I'm sure there's still a better *algorithm*.

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