The following finite state machine accepts all those binary strings in which the number of $$1's$$ and $$0's$$ are respectively

Which of the following grammar rules violate the requirements of an operator grammar ? $$P,$$ $$Q, R$$ are non-terminals and $$r, s, t$$ are terminals. $$$\eqalign{ & 1)\,\,\,P \to Q\,R\,\,\,\,\,2)\,\,\,P \to Q\,s\,R \cr & 3)\,\,\,P \to c\,\,\,\,\,\,\,\,\,\,\,4)P \to Q\,t\,R\,r \cr} $$$

Consider the following grammar $$G:$$
$$\eqalign{ & S \to bS\,\left| {\,aA\,\left| {\,b} \right.} \right. \cr & A \to bA\,\left| {\,aB} \right. \cr & B \to bB\,\left| {\,aS\,\left| {\,a} \right.} \right. \cr} $$

Let $${N_a}\left( w \right)$$ and $${N_b}\left( w \right)$$ denote the number of $$a's$$ and $$b's$$ in a string $$w$$ respectively. The language
$$L\left( G \right)\,\,\, \subseteq \left\{ {a,b} \right\} + $$ generated by $$G$$ is

The language $$\left\{ {{a^m}{b^n}{c^{m + n}}\left| {m,n \ge } \right.} \right\}$$ is