"Log" the both side: $$a\ln a+b\ln b+c\ln c\geq a\ln c+b\ln a+c\ln b.$$Since $(a,b,c)$ and $(\ln a,\ln b,\ln c)$ is up-up when $a\leq b\leq c,$ it is "Rearrangement". QED Grotex.

Yes this is the official solution @above

For any different positive numbers $a,b,c$ prove the inequality $$a^ab^bc^c>a^bb^cc^a.$$