Inclusion exclusion: How many bit strings of length eight do not contain six consecutive $0$'s? 
Q. How many bit strings of length eight do not contain six consecutive $0$s?

I solved this problem with my hand.
$$256 - ( 1 + 2 + 2 + 1 + 2 ) = 248$$
I calculated all possible events.
Is my answer right?
And can this problem be solved by the inclusion-exclusion principle? 

I think your answer is wrong but the idea is good. The number to subtract can indeed be found using the inclusion-exclusion principle, for 3 sets. Can you explain why you subtracted those numbers ?
EDIT: Apologies - I was mistaken. See N. F. Taussig's answer and comment.
