"For such Questions where It is asked that "Some things are never together", Try to arrange the remaining things first and then in the spaces created between them, put the Things that should never be together."
It's given that exactly 4 zeros, it means we have 6 1's, So, first of all arrange the 1's, which can be arranged in any order therefore 1! (since all 1 are identical)
1 1 1 1 1 1 since no two 0's should be consecutive, we have total 7 space after arranging 1's, we can place the four 0's in those spaces.
so select 4 places from the 7 spaces for 4 0's (all 0's are identical so all the permutations will be same or you can say there will be only one way of permutation) = 7C4
Therefore total binary string = 1! * 7C4 = 35
