The game of Tic-Tac-Toe

O will win the game. The sixth mark was X in square 9.
The 7th mark must be placed in square 5 which is the win situation for both X and O. Hence, the 6th mark must be placed in a line already containing two of the opponents marks. There are two such possibilities - the 6th mark would have been either O in square 7 or X in square 9.
As we know both the players are intelligent enough, the 6th mark could not be O in square 7. Instead, he would have placed O in square 5 and would have won.
Hence, the sixth mark must be X placed in square 9. And the seventh mark will be O. Thus O will win the game.
Answer
Statement (4) is false. There are 3 men, 8 women and 6 children.
Assume that Statements (4), (5) and (6) are all true. Then, Statement (1) is false. But then Statement (2) and (3) both can not be true. Thus, contradictory to the fact that exactly one statement is false.
So Statement (4) or Statement (5) or Statement (6) is false. Also, Statements (1), (2) and (3) all are true.
From (1) and (2), there are 11 men and women. Then from (3), there are 2 possible cases - either there are 8 men and 3 women or there are 3 men and 8 women.
If there are 8 men and 3 women, then there is 1 child. Then Statements (4) and (5) both are false, which is not possible.
Hence, there are 3 men, 8 women and 6 children. Statement (4) is false.