Answer

Correct Option: (iii) there exists a sentence with more than one derivation tree

Explanation

Ambiguous Grammar: Grammar $G$ is ambiguous iff there exists a string $w \in L(G)$ that has two or more distinct derivation trees (or equivalently two distinct leftmost / rightmost derivations).

• (i) “two productions have same LHS” → false; e.g., $S \rightarrow a \mid b$ has same LHS but not ambiguous (each string one tree).
• (ii) “derivation tree has more than one sentence” — every tree yields exactly one sentence (its yield), false.
• (iii) True — definition of ambiguity: $\exists w: |\{ \text{parse trees for }w\}| \ge 2$.
• (iv) “brackets not present” — irrelevant.

Example ambiguous: $S \rightarrow S+S \mid S*S \mid a$; string $a+a*a$ has two trees (different precedence).

Final Answer

Ambiguous iff some sentence has ≥2 parse trees → Option (iii).