$wpdb: Counting posts corresponding to 3 terms in 3 different taxonomies

Do a tax-query and then count the result. No need for a custom query with dozens of JOINS.

$posts = new WP_Query( array(
    'post_type' => 'MYPOSTTYPE', 
    'tax_query' => array(     
        array(         
            'taxonomy' => 'TAX_1',
            'field'    => 'slug',         
            'terms'    => array( 'TERM_TAX_1' ),
            'operator' => 'IN'
        ),
        array(
            // etc.
        )
    ),
    'post_status' => 'publish'
);
prinft(
     '<h3>COUNT: %s</h3>'
    ,count( $posts )
);