The other answers look good if formulae is guaranteed to exist on at most 1 line at a time. If it can appear twice on the same line, this will work.

use strict;
my $i;
while (<DATA>) {
  my @m = ($_ =~ /formulae/g);
  $i += @m;
}
print "Found $i matches\n";

__DATA__
message one
message two formulae formulae
message threee
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