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Given the following language, I am tasked with giving a context-free grammar that generates it.

$\left \{ a^{3n}b^{m}c^{n}|n>0, m>0\right \}$

Would this be correct?

$A \rightarrow aaaA$
$B\rightarrow Bb | b$
$C\rightarrow Cc|c$
$S\rightarrow ASBSCS|\epsilon $

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    – D.W.
    Nov 19 '20 at 19:26
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Unfortunately your grammar generates only $\epsilon$ (which is not in the language, since $n$ and $m$ are greater than 0), as you don't delete the $A$; moreover you don't control that the number of $A$'s is thrice the numeber of $C$'s.

You can try something like this:

$S\rightarrow aaaS'c$
$S' \rightarrow aaaS'c | B$
$B\rightarrow Bb | b$

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