Counting posts within categories

I think I’m on a wild track with a LEFT JOIN right?
I should maybe use sub selection. Something like

$querystr = "
		SELECT COUNT(*)
		FROM $wpdb->posts, $wpdb->term_taxonomy
		WHERE posts.post_type = 'post'
		AND posts.post_status = 'publish'
		AND ID = term_id
		AND term_id = 21;
		";
		$result = $wpdb->get_var($querystr);

After reading about the WordPress Taxanomy, I’m a bit uncertain if I should use wp_term_relationships or wp_term_taxonomy.

All I need is to count the number of published posts in a category. Shouldn’t bee that hard….

Finally, Thanks to all the help here I managed to solve it.
This code gets the number of published posts in one or more categories:

function get_post_count($categories) {
	global $wpdb;
	$post_count = 0;

		foreach($categories as $cat) :
			$querystr = "
				SELECT count
				FROM $wpdb->term_taxonomy, $wpdb->posts, $wpdb->term_relationships
				WHERE $wpdb->posts.ID = $wpdb->term_relationships.object_id
				AND $wpdb->term_relationships.term_taxonomy_id = $wpdb->term_taxonomy.term_taxonomy_id
				AND $wpdb->term_taxonomy.term_id = $cat
				AND $wpdb->posts.post_status = 'publish'
			";
			$result = $wpdb->get_var($querystr);
  		$post_count += $result;
   endforeach; 

   return $post_count;
}

I also tested trying to count unpublished posts. That however didn’t work. According to Writing Posts, a post has three states; Published, Pending Review, and Unpublished. But looking at other code I find the following post_status: draft, future, publish and private.
Changing the $wpdb->posts.post_status = ‘publish’ to any of the other three, only returns 0 (zero) even though I have created 2 posts and not published them. So I have no clue on how to count those.

And if you try to use the wp_count_posts(), you will not get the number of posts for a specific category. I’m not quite sure what it actually reads.