There is a runner on third with two out. Batter hits a slow roller to first base. First baseman charges and picks up the ball. Runner from third crosses home. First baseman tags the runner running from home to first, does the run count ?
The run does not count. According to MLB rules:
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.
Since the batter-runner does not make it to first base safely, the run does not count. Similar to a force out.