What is the Correct SQL - Need to Join 3 Tables

[Stefanie Steckerl]

I have 3 tables: wp_categories, wp_post2cat and wp_posts. I am trying to display the category names (cat_name) from wp_categories for each distinct occurrence of category_id from wp_post2cat where the value mem_level is <= 1 and the post_status is = "publish" from the wp_posts table. In wp_categories, the primary key is cat_ID for each cat_name. The primary key in wp_post2cat is post_id. In wp_posts, the primary key is ID. This table contains all the posts with the values for mem_level as well as post_status. The other thing I need to do is display a count of the number of distinct category_ids in wp_post2cat where the value mem_level is <= 1 and the post_status is = "publish" in the wp_posts table. I thought the following SQL might work, however, it doesn't bring back any results - can anyone help me with this?:
SELECT cat_name,
COUNT(post2cat.category_id) AS cat_count
FROM wp_categories
INNER JOIN post2cat ON (cat_ID = category_id)
INNER JOIN wp_posts ON (ID = post_id)
WHERE post_status = 'publish'
AND post_date_gmt < '$now' $exclusions
GROUP BY category_id
Any help would be greatly appreciated. Thank you!