Thanks so much for your help, stvwlf!
In case anyone needs to do this, I modified the code from here for multiple posts.
$post_limit_per_cat = 5;
$post_ids = array(0);
foreach( array(7) as $cat_id ) {
if ( $posts = get_posts(array('cat' => $cat_id, 'showposts' => $post_limit_per_cat)) ) {
$post_current_count = 0;
while( (count($posts) > 0) AND ($post_current_count < $post_limit_per_cat) ) {
$first = array_shift($posts);
$post_ids[] = $first->ID;
$post_current_count++;
}
}
}
query_posts(array('post__in' => $post_ids));