context free languages under SCRAMBLE operation

A question from a practice exam found online...(also in Sipser, I believe)

For strings w and t, write w =' t if the symbols of w are a permutation of the symbols of t.
In other words, w =' t if t and w have the same symbols in the same quantities, but possibly
in a different order.
For any string w, define SCRAMBLE(w) = {t | t =' w}. For any language A, let SCRAMBLE(A) = {t | ϵ SCRAMBLE(w) for some w ϵ A}.
a. Prove that, if Σ = {0, 1}, then the SCRAMBLE of a regular language is context free.
b. What happens in part (a) if Σ contains 3 or more symbols? Prove your answer.



For part (a) I think this can be proven by simply noting that the DFA
which accepts the regular language to be scrambled can simply have its state transitions re-arranged to accept the SCRAMBLE of the language.
This is de facto context free since every regular language is in turn context free.
Am I right?
For part (b) if the language of (a) has 3 or more symbols than it is
not context free, I believe, however I am not sure how to prove this?
Perhaps a counter example? How about this counter example,
The language containing strings like
cabcab
is obviously regular, however we know from a previous result that
strings of the form a^nb^nc^n n>=0 are not context free.
Is that a sufficient proof?
.

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