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How to display random posts from a given category? »

How do I show latest post from X random categories?

<?php
	// grab 4 random categories
	$cats 		= '';
	$categories	= get_categories();
	$rand_keys 	= array_rand($categories, 4);
	foreach ($rand_keys as $key) {
		$cat = $categories[$key]->term_id;
		$nam = $categories[$key]->cat_name;

	// grab the latest post from each of the 4 random categories
	$latest = new WP_Query("cat=$cat&amp;showposts=1"); 

	while($latest->have_posts()) : $latest->the_post();
?>

<small>Latest in <a href="#"><?php echo $nam; ?></a></small>
<h3><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></h3>
<?php the_excerpt(); ?>

<?php endwhile; ?>
<?php } ?>