how to get number of posts in a category on a specific date

You would want to use something like this:

$query = new WP_Query( 'year=2013&monthnum=12&day=20&cat=5' ); 
// change cat=5 to the ID of the category you want to query from.
// change year to the year you want
// change monthnum to the month you want
// change day to the day you want

// start the loop
while( $query->have_posts() ) : $query->the_post();
     // do loop stuff
endwhile;

wp_reset_postdata();

Above says get all posts that has ID of 5 for category and the year of 2013 and month is 12 (December) and day is 20.