I have $2m$ student, $m$ girls and $m$ boys, and I want to order them in $k$ classes with $m$ capacity of each class, so there will be at least one girl in each class. I calculate it by inclusion-exclusion principle:
$$w(0) = P_{km}^{2m}$$
$P_i\:-\text{in the i class there isn't a girl}$
So, as I understand, for $w(r)$ I choose $\binom kr$ classes that will not have a girl, then $P_{m\left(k-r\right)}^m$ ordering the girls in all other classes, then ordering the boys in all classes $P_{mk}^m$ So: $$w\left(r\right)=\binom krP_{m\left(k-r\right)}^mP_{mk}^m$$
But the right answer for $w(r)$ is: $$w\left(r\right)=\binom krP_{m\left(k-r\right)}^mP_{m\left(k-1\right)}^m$$ I don't see why it should be $m(k-1)$ in the last expression. Generally, the next step is:$$E\left(0\right)=\left(-1\right)^rw\left(r\right)$$ but that isn't the issue. Cam somebody explain it to me please?
Your notation for permutations is also confusing (to me), so instead of confusing you further with my notation, I'll just explain simply.
You are seating all the $m$ girls first, so seats remaining in which boys can be permuted is $mk - m = m(k-1)$