CFG for palindromes containing atleast 3 c's
You have a symbol S that needs to generate a palindrome with at least 3 c's.
Create a symbol X that needs to generate a palindrome with at least 1 c, and a symbol Y that needs to generate a palindrome with no requirement for c's. 

S -&gt; aSa | bSb | cXc
X -&gt; aXa | bXb | cYc | c
Y -&gt; aYa | bYb | cYc | a | b | c | eps


I think the essential step was not to start writing down productions, but to write down which different symbols you need to produce which kinds of output. The productions are then trivial.