Consider the following statements in respect of the events A , B ,...

Using De Morgan's laws, we have $\overline{A}\cap\overline{B}\cap\overline{C} = \overline{A\cup B\cup C}$.

Substituting this in the expression for Statement I:
$(A\cup B\cup C)\cap(\overline{A}\cap\overline{B}\cap\overline{C}) = (A\cup B\cup C)\cap\overline{(A\cup B\cup C)} = \phi$

Since the intersection of an event and its complement is the empty set $\phi$, it represents an impossible event. Thus, Statement I is correct.

Again, using De Morgan's laws, we have $\overline{A}\cup\overline{B}\cup\overline{C} = \overline{A\cap B\cap C}$.

Substituting this in the expression for Statement II:
$(A\cap B\cap C)\cap(\overline{A}\cup\overline{B}\cup\overline{C}) = (A\cap B\cap C)\cap\overline{(A\cap B\cap C)} = \phi$

This is also an impossible event. Thus, Statement II is incorrect.

Therefore, only Statement I is correct.