Which word to pump in pumping lemma?
You can use any word in the language that lets you complete a proof. The pumping lemma says that, if a language&nbsp;$L$ is regular/context-free, then every long-enough string in the language can be partitioned in a way that generates an infinite set of strings that are also in&nbsp;$L$. So, if you can find any string in&nbsp;$L$ where the partition generates strings outside&nbsp;$L$, you're done.

Finding the right string to start with requires some thought and creativity; I'm not aware of any general method for doing it. I've not written anything out but I think that, if you start with $0^{n+1}1^n$, the proof won't work. However, you should try doing the proof with that string. Assuming that you don't make any mistakes, there are two possible outcomes:


you win by successfully proving that the language isn't regular;
you find that the proof doesn't work, and if the reason is the one I'm expecting, it should guide you straight to a better string to try to pump.