Help generating a grammar with {a,b,c} where the number of as and bs are different or bs and cs, or both w ∈ C*
"Generate a Grammar such that w contains strings of as followed by bs followed by cs such that there are either a different number of as and bs or a different number of bs and cs or both"
C= {a,b,c}
I am trying to do both. I'm having difficulties generating grammars, but I have understood that I should just write down some strings and from there try to generate some rules.
some of my strings are aabc, abbc, abcc, aabbbcc
I have produced the following rules:
$S\rightarrow aSbScS$
$S\rightarrow AS$
$S\rightarrow SB$
$S\rightarrow SC$
$S\rightarrow e$
$A\rightarrow a$
$B\rightarrow b$
$C\rightarrow c$
Did I do this right? And if not, what did I do wrong? 
I am a language student and am trying to make my brain get used to such thinking, but I am somehow struggling with this.
Thank you.
 A: Look at it like this, at least one of the following has to be true:


*

*the number of $a$'s is larger than the number of $b$'s

*the number of $b$'s is larger than the number of $a$'s

*the number of $b$'s is larger than the number of $c$'s

*the number of $c$'s is larger than the number of $b$'s


So we can make a grammar for each of these, and then simply combine them.
A grammar for the first condition looks like this:
\begin{align*}
S_1&\to aXC\\
X&\to aX \mid aXb\mid\varepsilon\\
C&\to cC\mid\varepsilon
\end{align*}
The first rule means there is at least one $a$ in the string, which is necessary.
The second rule means that either we can add $a$'s to the front, or we can add both an $a$ to the front and a $b$ to the middle. Finally, the number of $c$'s doesn't matter.
Now we can make similar grammars with starting variables $S_2$, $S_3$ and $S_4$ for the other three conditions above, and then we can combine these with the rule
\begin{align*}
S\to S_1\mid S_2\mid S_3\mid S_4
\end{align*}
