The original question is given in an exercise. The first part of the problem is to show that if $\left|A\right| = \left|A\right| +\left|A\right|$ for some set $A$, then there exists a partition of $A$, with every cell in the partition having cardinality $2$ (i.e. a partition of $A$ into pairs). The second part of the problem is to show that the cardinality of the set of permutations of $A$ is $2^{\left|A\right|}$.
Using the first part of the problem, it's not hard to show that that cardinality of set of permutations of $A$ is bounded between $2^{\left|A\right|}$ and $\left|A\right|^{\left|A\right|}$. But I couldn't find a way to proceed from here without assuming that $\left|A\right| = \left|A\right|\cdot\left|A\right|$.
Is it possible to complete the proof without choice?