There are $4$ different types of balls, and $5$ bins. The bins are labeled $A,B,C,D,E$. Each bin only holds one ball. We are to put the balls into each bin such that every type of balls is put into at least one bin.
Repetition is allowed, and you have an infinite supply of each type of balls. Condition: the balls in $A$ and $B$ must be of different type. How many different ways are there to do this?
After filling $A$ and $B$, there are only $2$ types of balls left but $3$ bins. So the last bin can hold any type of ball. There are $4$ ways to fill $A$, $3$ to fill $B$, $2$ to fill $C$ and $1$ way to fill $D$. Finally, there are $4$ ways to fill $E$, the last bin.
So, in total, there are $4(3)(2)(1)(4) = 96$ ways to do this.
Is my reasoning correct?
Is there a more systematic way to tackle similar questions about permutations with repetition and conditions? Or is my way the most systematic?