Does the run count?

[Rich Ives]

Correct.

2.00 A FORCE PLAY is a play in which a runner legally loses his right to occupy a base by reason of the batter becoming a runner.

4.09 HOW A TEAM SCORES.
a) One run shall be scored each time a runner legally advances to and touches first, second, third and home base before three men are put out to end the inning. EXCEPTION: A run is not scored if the runner advances to home base during a play in which the third out is made (1) by the batter runner before he touches first base; (2) by any runner being forced out; or (3) by a preceding runner who is declared out because he failed to touch one of the bases.