Let me edit your code, which is very unsafe once the CSV gets a little bit more complicated ...

use Text::CSV_XS;

my $csv = Text::CSV_XS->new ({ binary => 1, auto_diag => 1 });
open my $data, "<", $file_c or die "$file_c: $!\n";
my $rows = 0;
while (my $row = $csv->getline ($data)) {
    $rows++;
    }
print "Total $rows rows (including header line)\n";
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Though I'd probably go to an even simpler solution using DBD::CSV:

$ ll xx.csv
12862169 -rw-rw-rw- 1 merijn users 415423 Sep  6 16:00 xx.csv
$ perl -MDBI -wE'say DBI->connect("dbi:CSV:f_ext=.csv")->selectrow_arr
+ayref("select count(*) from xx")->[0]," rows in fs"'
8739 rows in fs
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