How many ways to assign $8$ teachers?

Question

$8$ new teachers are going to be sent to $4$ schools.

This is a 2 part question.

How many assignments are possible? Here the answer is $65\ 536$, but I don't understand how they got that answer.

What if every school receive exactly $2$ teachers? Here my intuition would have been to proceed like this:

$\binom{8}{2}\cdot 4!$

But the correct answer is this:

$\binom{8}{2}\binom{6}{2}\binom{4}{2}\binom{2}{2}=2520$

so if anyone could enlighten me it would help me alot. Thank you.

Answer 1

For second part, the formula given simply indicates that you choose any 2 teachers for the 1st school, 2 from the remaining 6 for the 2nd school, and so on.

It may help you to realize that the formula can also be written as:

$\dfrac{8!}{(2!)^4}$, which means that we permute all the 8 teachers, and remove permutations between teachers in each school.

And if you have studied the multinomial coefficient, it could also be written as $\dbinom{8}{2,2,2,2}$

Note that each school has been considered to be distinct