3rd out run

Question
Here is the scenario.  Man on third base, two outs.  Batter hits a ground ball to the pitcher.  Runner on third runs to home.  Pither throws to first.  First baseman drops the ball.  While ball is on the ground, the runner scores.  First baseman then picks up the ball just in time to get the batter out for the third out.  Does the run count?  Thank you.

Answer
No, the run would not count because the out at first was a force play for the 3rd out. Now, let's say the out at first wasn't made until the batter-runner went from first and got tagged out when he tried to advance to 2nd. In that case, the run would score.