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F A L S E.
The correct answer is 6*5*4*3 = 360 ways.
Assuming, by default, that every of four classes obtains an assigned teacher.
It is of the same type problem as another one (a TWIN) solved under this link
https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1173428.html
https://www.algebra.com/algebra/homework/Permutations/Permutations.faq.question.1173428.html
The same mantra works.