If you want the number of descendants, you can use
var element = document.getElementById("theElementId");
var numberOfChildren = element.getElementsByTagName('*').length
But if you want the number of immediate children, use
element.childElementCount
See browser support here: http://help.dottoro.com/ljsfamht.php
or
element.children.length
See browser support here: https://developer.mozilla.org/en-US/docs/DOM/Element.children#Browser_compatibility