# Which part of the string $00011112222$ can be pumped?

Let us observe the language $$\newcommand{\lang}{\mathcal L} \lang = \newcommand{\set}[1]{\left\{ #1 \right\}}\set{ 0^i 1^n 2^n \mid i \geq 1, n \in \mathbb{N}}$$. Can any of the substrings indicated with $$[\ldots]$$ below be pumped using the pumping lemma for context free languages?

1. $$00[0]11112222$$
2. $$00[01]1112222$$
3. $$00011[1122]22$$

The pumping lemma for context free languages states, that if a language $$\lang$$ is context free, after some limiting length $$\ell$$ any string $$x \in \lang$$ should be partitionable into the string $$x =x_1 x_2 x_3 x_4 x_5$$, where $$x_2x_4 \neq \epsilon$$ and $$\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}\abs{x_2x_3x_4} \leq \ell$$, so that $$x^\prime = x_1x_2^kx_3x_4^kx_5 \in \lang$$ for any $$k\geq 0$$ as well. The repetition of the strings $$x_2$$ and $$x_4$$ in the middle of the string $$x$$ is called pumping.

Because of this:

1. is not an option, as $$[0] = [x_2x_3x_4]$$ would mean that either $$x_2$$ or $$x_4$$ is empty.

2. This does not work either, as $$[01] = [x_2x_3x_4]$$ would mean that for $$k\neq 1$$, the pumped word would no longer be in the language. This is because the number of $$1$$s would either exceed or subceed the number of $$2$$s.

3. This is a bit trickier, as with $$[1122] = [x_2x_3 x_4]$$ we could choose $$x_2=11$$, $$x_3 = \epsilon$$ and $$x_4 = 22$$. Now pumping the strings $$x_2$$ and $$x_4$$ would still have the word $$x^\prime$$ in the language: the increase (or decrease) of the number of $$1$$s is always matched by the changes in the number of $$2$$s. So this seems to match the requirements for pumping.

However, according to a certain question submit form, my answer is incorrect and option 3 cannot be pumped at the indicated location. Why is that? Does it have something to do with the pumping length $$\ell$$ of the word $$x$$?