You can use (?{...}) (see perreref's EXTENDED CONSTRUCTS) to execute a code fragment inside a regex so:

while (<IN>) {
    if (m/^\d+\sof\s\d+\sDOCUMENTS$/mx ... m/^No.\s\w+$/mx){
        $casename = $1 if /^(.*)/;
        /EDUCATION(?{$count++})/g;
    }
}
[download]

would seem to do what you want.