Creating archive pages for children categories

Example:

video cat id = 4
video cat child category id’s = 7,8,9
news cat id = 5
news cat child category id’s = 11,12,13

I wanna display choosen categories ( like video and news ) in the different archive.php?

Is it possible thanks.

( sorry about my bad Eng )

Thanks for Chris S but its not what i want to do.

  <?php
$catPosts = new WP_Query();
$catPosts->query( array( 'category__and' => array(5,11,12,13), 'posts_per_page' => 5, 'orderby' => 'date', 'order' => 'DESC' ) );
  while ($catPosts->have_posts()) : $catPosts->the_post(); ?>
  <h1><a href="https://wordpress.stackexchange.com/questions/24423/<?php the_permalink() ?>"><?php the_title(); ?></a></h1>
  <div class="meta">
   By <?php the_author() ?>
  </div>
<div class="content">
  <?php the_excerpt(); ?>
</div>

When I wrote the parent category id, i dont wanna write its child category id’s. When i choose the parent category id the system should its child category ids in the same archive page.

Create 2 pages category-news.php & category-video.php & put this code in them. Then customize the markup for both as you like

<?php
$children = get_categories('child_of'=>get_query_var('cat'));
$cat = array(get_query_var('cat'));
foreach($children as $child)
    $cat[] = $child->term_id;
$catPosts = new WP_Query( array( 'category__in' => $cat, 'posts_per_page' => 5, 'orderby' => 'date', 'order' => 'DESC' ) );
  while ($catPosts->have_posts()) : $catPosts->the_post(); ?>
  <h1><a href="https://wordpress.stackexchange.com/questions/24423/<?php the_permalink() ?>"><?php the_title(); ?></a></h1>
  <div class="meta">
   By <?php the_author() ?>
  </div>
<div class="content">
  <?php the_excerpt(); ?>
</div>