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Just count the spaces and add 1:

// No need to count markup.
$text        = trim( strip_tags( get_the_content() ) );
$word_number = substr_count( $text, ' ' ) + 1;

Another way, taking the old American double space tradition into account:

$word_number = preg_match_all( '~\s+~', $text, $m ) + 1;