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Question
bases loaded with one out.  fly ball is hit to right. Ball is caught. two outs.  Runners tag up.  Runner on third scores, runner on second advances and attempts to score on a throwing error.  Runner on first attempts to advance also and is tagged out before the runner from second can cross the plate. umpire rules that tag was before the run and the second run does not count.  Defense makes dead ball challenge that the runner on second left before the catch in right. umpire agrees.  The first run is allowed to count.  Defense wants no runs to count.  umpires confirms that first run counts.  Should it count?

Answer
Yes the run counts.  

The common perception is that the appeal for the out at second is a force out, and if it is the third out no runs can score.  In reality it is not a force out, it is an out on appeal.  This makes it a timing play, just like the runner on first who was tagged out for the third out.  So, since the run scored before the third out was made, run scores.

Tom