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I want to design a context-free grammar that generates words that either both start and end with $c$, or contain the same amount of $a$-s and $b$-s. Here is what I have. The nonterminals are $S,X,Y$, the terminals are $a,b,c$, the start symbol is $S$, and the productions are

\begin{align} &S \to aYS \mid cXc \mid bXS \mid cYc \mid \varepsilon \\ &X \to a \mid bXX \\ &Y \to b \mid aYY \\ \end{align}

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Your grammar doesn't seem to generate $caac$. Indeed, in order to generate this work, a simple case analysis shows that the derivation must start $$ S \to cYc \to caYYc, $$ at which point it is clear that we cannot generate $caac$.

I assume you know how to generate the language $c\Sigma^*c$, as well as the language of all words with an equal number of $a$-s and $b$-s. Instead of combining them in a complicated way, write grammars for both, and then join them together using a rule of the sort $S \to S_1 \cup S_2$.

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