The answer I got is:

Here's the code I used:

use List::Util 'min';
use POSIX 'ceil';

sub num_ways {
    my ($N, $S, $T) = @_;

    return 1 if $S == 0 and $T == 0;
    return 0 if $N < 1 or $S < 1 or $T < 1;

    my $min = (2*$T-$S+$S**2)/(2*$S); ## more correctly, ceil() of thi
+s
    my $max = min( $T-(($S-1)*$S/2), $N );

    my $sum = 0;
    for my $K ($min .. $max) {
        $sum += num_ways( $K-1, $S-1, $T-$K );
    }
    return $sum;
}

use Memoize;
memoize 'num_ways';
print num_ways(100, 10, 667), $/;
[download]

num_ways($N,$S,$T) computes the number of ways to pick a S (distinct) elements from the range {1..N} whose sum is T. Here's how I compute it recursively:

The code can be made to not only count the solutions, but to iterate over them as well (using a callback & accumulation of the solution):

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