Consider the following deterministic finite automaton (DFA) The number of…

Key insight: after reading three symbols the automaton reaches an accepting sink state, so every binary string of length at least 3 is accepted.

• Step 1: The first input symbol sends the machine to one of the two left-middle states (0 → upper, 1 → lower).

• Step 2: The second symbol moves the machine to one of the two right-middle states determined solely by that second symbol (0 → top-right, 1 → bottom-right).

• Step 3: The third symbol always leads from either right-middle state to the accepting sink state, which has loops on both 0 and 1. Thus the machine stays in an accepting state for all remaining symbols.

Conclusion: Every binary string of length 8 is accepted, so the number of accepted strings of length 8 is 2^8 = 256.