$13$

The following question is given: Use the Pumping Lemma with length to prove that the following language is nonregular:

$L=\{ba{b}^{n-2}{a}^n,$ with nin$\{3,4,5,$dots$\},$ninZ$\}.$

The solution to this question is partly given as follows:

Assume $L=\{\begin{array}{c}ba{b}^{n-2}{a}^n\end{array},$ where $\{$:nin$\{3,4,5,$dots$\},$ninZ$\}$ is regular.

Then there exists an FA with, say, $k$ states, that accepts $L.$

Let $w=ba{b}^{k-2}{a}^k$ be a word in $L.$

According to the pumping lemma, $w$ may be written as

$w=xyz$ such that

length $(x)+$ length $(y)k$ AND length $(y)>0$

Which one of the following is not one of the possible correct choices for $y?$

$y$ comprises the $a$ in between $b$ and $b^{k-2}.$

$y$ comprises the first ba$-$substring.

$y$ comprises ba followed by possible $b\text{'}$s$.$

$y$ comprises ba followed by a possible second ba substring.