Which one of the following regular expressions is equivalent to the language…

Key insight: the DFA toggles state on each 1 and stays in the same state on 0, so it accepts exactly those strings over {0,1} with an odd number of 1s.

• Interpretation of states: start state represents an even number of 1s seen so far (initially 0). Each 1 flips parity; each 0 leaves parity unchanged. Acceptance occurs when the parity is odd.

• A compact regular expression for strings with an odd number of 1s is 0* (1 0* 1 0*)* 1 0*: initial zeros, any number of pairs of 1s (each pair contributes two 1s, keeping parity), then a final single 1 and trailing zeros.

• Equivalence to the correct option in the question: the expression 0*1(0 + 10*1)* is equivalent because after the initial 1 each repetition either adds a 0 (no new 1) or adds a 10*1 (two additional 1s). Thus these repetitions preserve odd parity of 1s and generate exactly the same set of strings as the canonical form above.