Case 1: Exactly one block of five consecutive 0’s.
The block $00000$ can start at positions 1 to 6, so 6 choices.
The remaining 5 positions can be filled freely with 0 or 1, except they should not create another block of five 0’s.
Valid fillings = $2^5 - 1 = 31$.
So number of strings with exactly one block of five 0’s is
$6 \times 31 = 186$

Case 2: Exactly one block of five consecutive 1’s.
By symmetry, the count is the same.
$186$

Case 3: One block of five 0’s and one block of five 1’s.
This is possible only when the blocks do not overlap.
The only such strings are
$0000011111$ and $1111100000$

So total such strings = $2$.
Using inclusion–exclusion principle:
$186 + 186 + 2 = 222$
Final Answer: $222$