Consider the following two languages over the alphabet $\{a, b, c\}$, where $m$ and $n$ are natural numbers.

$$\begin{aligned} & L_1=\left\{a^m b^m c^{m+n} \mid m, n \geq 1\right\} \\ & L_2=\left\{a^m b^n c^{m+n} \mid m, n \geq 1\right\} \end{aligned}$$

Which ONE of the following statements is CORRECT?

Consider a context-free grammar $G$ with the following 3 rules.

$S \rightarrow aS, \ S \rightarrow aSbS, S \rightarrow c$

Let $w \in L(G)$.

Let $n_a(w)$, $n_b(w)$, $n_c(w)$ denote the number of times $a$, $b$, $c$ occur in $w$, respectively. Which of the following statements is/are TRUE?

Let G = (V, Σ, S, P) be a context-free grammar in Chomsky Normal Form with Σ = { a, b, c } and V containing 10 variable symbols including the start symbol S. The string w = a³⁰b³⁰c³⁰ is derivable from S. The number of steps (application of rules) in the derivation S ⟹ w is _______

Consider the context-free grammar G below

$$\matrix{ S & \to & {aSb|X} \cr X & \to & {aX|Xb|a|b,} \cr } $$

where S and X are non-terminals, and a and b are terminal symbols. The starting non-terminal is S.

Which one of the following statements is CORRECT?