I'm having trouble constructing a Context Free Grammar for the following language: $$a^{\ast}b^{\ast}c^{\ast} - \{a^{n} b^{n} c^{n} \mid n \geq 0 \}$$I believe it's telling me that no string can be generated that has the same number of of $a$'s $b$'s and $c$'s.  This is due to the subtraction of the second set.  A good string from a newly formed language should be something like $aaabbc$ or $abbbcc$ and so on.So I tried breaking the problem into three parts...Single states:$S_{1} \rightarrow aS_{1} \mid a \mid \lambda$$S_{2} \rightarrow bS_{2} \mid b \mid \lambda$$S_{3} \rightarrow cS_{2} \mid b \mid \lambda$Two States$S_{4} \rightarrow aS_{4}b \mid S_{1} \mid S_{2}$$S_{5} \rightarrow bS_{5}c \mid S_{2}$$S_{6} \rightarrow aS_{6}c \mid S_{3} \mid S_{1}$States w/AB states$S_{7} \rightarrow S_{1} \mid S_{4}S_{6}$$S_{8} \rightarrow S_{2} \mid S_{5}S_{6}$$S_{9} \rightarrow S_{3} \mid S_{6}S_{3}$with an orginal start state of   ...$$S \rightarrow  S_{7} \mid S_{8} \mid S_{9}$$However I'm having problems building strings like $aaaabbbcc$ ...Am I forming CFG's incorrectly?  I felt like I was on the right track but now I'm quite lost.  
Context Free Grammar for $a^*b^*c^* - \{a^n b^n c^n \mid n \geq 0 \}$
formal languages;context free;formal grammars