What language does this mean?

Question

this is the language:

L = { w belong {a,b,c}* | |w|= 3 * number(a) (w) }

Then, what does that mean?

Answer 1

It means that L is the language of strings w consisting of symbols 'a', 'b'' and 'c', where the length of the string w equals to 3 times the number of symbol 'a' present in the string w.

The productions for this grammars should be such that if it add one 'a' then it also adds two 'b', or two 'c', or one 'b'; one 'c'. Check below grammar:

S → ^ | SaSMSM |  SMSaSM | SMSMSa   
M → b | c

here ^ means epsilon.

To generate aabbcc use Right most derivation

1. S → SaSMSM
2. replace first S in rhs by ^ using S → ^
   S → SaSMSM → aSMSM
3. replace S → SaSMSM
   S → SaSMSM → aSaSMSMMSM
4. use S → ^
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM
5. use S → ^
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM
6. M → b
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM
7. use S → ^
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM
8. M → b
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM
9. M → c
   S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM → aabbcSM
10. use S → ^
    S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM → aabbcSM → aabbcM
11. M → c
    S → SaSMSM → aSaSMSMMSM → aaSMSMMSM → aaMSMMSM → aabSMMSM → aabMMSM → aabbMSM → aabbcSM → aabbcM → aabbcc

Answer 2

S -> aBCS | aBC
S -> BCaS | BCa
S -> BaCS | BaC
S -> aCBS | aCB
S -> CaBS | CaB
S -> CBaS | CBa
B -> b
C -> c

may be right ?