Let $P$ be the finite-support product of the Cohen forcing. It adjoins a sequence of Cohen-generic reals say $a_n$, $n<\omega$, which one naturally calls *a $P$-generic sequence*. Suppose that $\pi$ is a permutation of $\omega$. Consider the new sequence of reals $a'_n=a_{\pi(n)}$.

Will the new sequence be $P$-generic?

It's answered in the positive provided $\pi$ belongs to the ground model $M$.

Now, what about the case $\pi\notin M$, but $\pi$ belongs to the extension $M[(a_n)_{n<\omega}]$?