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.