The split solution is not correct, because it does not account for multiple occurences of $pattern at the end of the string:

$str = 'ababbb';
$pattern = 'b';
$count = (scalar split /$pattern/, $str )
         + ($str=~/$pattern$/)
         - 1;
print "$count\n";
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2
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Fortunately, this is an easy problem to fix:

$count = (scalar split /$pattern/, $str, -1) - 1;
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The third argument to split specifies the maximum number of pieces to split the string into. A negative number turns off the stripping of null fields from the end of the list, without limiting the number of pieces.