WordPress query for most recent posts from multiple categories

I have 5 categories. I want to query each category and get a list of the most recent posts from them. I also only want single post from each category, with no duplicates. Is there a way to do this without writing 5 queries? Can it be done with one query?

I should end up with the following results:

1|most recent post from CategoryA
2|most recent post from CategoryB
3|most recent post from CategoryC
4|most recent post from CategoryD
5|most recent post from CategoryE

Here is some code I used for the front page of this site. It pulls one post (I chose a random post) from each category and displays some of the post content. I’ve included the HTML so you can make sense of it.

<div id="main-content" class="main-content-wide">    
<?php
    //Get the desired categories and order by ID
    $cat_args = array(
    'include' => '7,8,9,10, 14, 60',
    'orderby' => 'ID',
    'child_of' => 0
    );

//For each category show a random post
$categories =   get_categories($cat_args); 
foreach($categories as $category) 
    {
?>        

<div class="categorytitle">    
    <h3>Children's Stories <?php echo $category->name; ?></h3>
    <a href="https://wordpress.stackexchange.com/questions/129022/<?php echo get_category_link( $category->term_id ); ?>" class="allstories"> (More Stories...)</a>
</div>

<div class="story">
    <?php
    $post_args = array(
    'numberposts'   => 1,
    'orderby'   => rand,
    'category'  => $category->term_id
    );

    $posts = get_posts($post_args);
    foreach($posts as $post)    
    {
?>    
//Show post content     
<?php get_template_part('loop'); ?>
</div>
<?php
        }
}
?>   

</div>