there is 2^8 strings of 8 bits.

there is 2^4 strings of 4 bits, so there is 2^4 strings of 8 bits that start with 0000 (each 8 bits string that start with 0000 is like a 4 bits string).

so there is 2^8-2^4+1 strings of length 8 without having at least 2 bit strings that start with 0000.