use List::Util 'sum';

sub fact {
    my $n = shift;
    my $f = 1;
    $f *= $_ for 1 .. $n;
    $f;
}

sub prob {
    my ($sizes, $sample) = @_;
    my $n = @$sizes;     ## number of types of items
    my $S = sum @$sizes; ## total number of items

    my @sampled = (0) x $n;
    $sampled[$_]++ for @$sample;
                         ## $sampled[x] = how many of type x
                         ##     are in this sample?

    my $total = fact(scalar @$sample);
    $total *= ( ($sizes->[$_] / $S) ** $sampled[$_] ) / fact($sampled[$_])
        for 0 .. $n-1;

    $total;
}

## how many ways to get 2 of type 8, 1 of type 3, etc.. ?

print prob( [35000,41000,16000,18000,21000,45000,27000,10000,16000],
            [8,8,3,1,5,0,0] );

## output: 0.000397344401565931