Uncaught 3rd Strike

Question
A runner is at third with 2 out. The count is 2 and 2. The batter swings and misses at the next pitch which is low and skips past the catcher. The ball recoils off the backstop to the catcher who lunged at the runner coming home. Not in time. SAFE. The batter got a late start out of the box and the catcher nips the runner at first with a great throw from his knees. Does the run count? The younger umpire behind the plate said no and the senior umpire on the field over ruled and said yes.

Answer
In this case, the younger umpire was right.  Dropped third strikes constitute a force play, meaning the runner must make it to first before any run would count, much the same way it would be if the kid had grounded out to short.  I hope this answers your question.