How can i list random post from multiple category?

Granted, I may be misinterpreting you. Your post is not very clear but …

category__and will require the posts to be in both categories. It sounds like you need posts in any of the categories. You should be using category__in so that you get an OR match on the categories instead of an AND match.

showposts is deprecated.

Both category__and and category__in accept an array. You are building a comma separated string of categories and then casting it as an array. That is not likely to work as expected.

Build your category array like this:

foreach($categories as $category) : 
    $catnumber[] = $category->ID;
endforeach;

And query like this:

$args = array(
    'category__in'    => $catnumber,
    'posts_per_page'  => 10,
    'orderby'         => 'rand'
);

http://codex.wordpress.org/Class_Reference/WP_Query#Category_Parameters