No, because the third out was made on the same play. It is no different than if the batter hits a ground ball to the shortstop or any other infielder and is thrown out at first base for the third out. However, if there are two outs and there is a runner on third and the batter hits a single into left field but is thrown out at second when attempting to turn the play into a double and the runner on third makes it home before the third out at second is recorded, the run does count. Hope that isn't too confusing.
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The only way that run would count is if it were not a continuous double play... For example, if the batter grounds into a routine double play, the run does not score... but if the batter were to fly out and another runner tries to tag up, and advance from 1st the second, or from 2nd to 3rd, and is thrown out in the process, as long as the runner for 3rd crosses home plate before the out is recorded, it counts.
To simplify the answer a bit... If the double play occurs with both outs being recorded on force outs, no run scores. If there are multiple outs recorded, but are not force outs, the run scores as long as the runner crosses before the out is recorded.
the run doesnt count because it depends on if the runner is safe. but if he gets out the runner does not get counted with the run.
yes, as long as there was only 0 or 1 out b/c if there was 2 outs the out on the advancing runner would end the inning