Question

If $8$new teachers are to be divided among $4$schools, how many divisions are possible? What if each school must receive $2$teachers?

Short answer

$\left(a\right)65530$-- the basic principle of counting

$\left(b\right)2520$- use the multinomials.

Step-by-step solution

Given Information.

Let $8$new teachers are to be divided among $4$schools.

Part (a) Explanation.

Divide $8$people among $4$schools

The teachers are different among themselves, and so are the schools.

Every teacher can choose a school. By the basic principle of counting:

localid="1649848745500" $4·4·4·4·4·4·4·4=4^8=65536$different divisions of the teachers.

Part (b) Explanation.

Every school gets preciseness $2$teachers:

To divide $8$different objects into four groups of fixed sizes$2,2,2$and2.

localid="1649848759459" $\left(\begin{array}{c}8\\ 2,2,2,2\end{array}\right)=\frac{8!}{2!2!2!2!}=2520$.different divisions of the teachers.