Suppose I throw 10 balls into 5 bins probability expectancy problem

Question

Suppose I throw 10 balls into 5 labeled boxes labeled A,B,C,D,E uniformly at random. Call a pair of these boxes "special" if they are adjacent(i.e. AB, BC, CD, DE). The expected number of special boxes which are BOTH empty is?

Answer 1

What is the probability a given "special box" is empty?

Let's look at the case $AB$. For $AB$ to be empty (correct me if I am interpreting the problem wrong) means that no balls fall into box $A$ or $B$. Meaning all balls fall into $C$, $D$, or $E$.

The probability of this occurring is $\frac35^{10}$ (each of the 10 balls having a $\frac35$ chance of landing in a given box $C$, $D$, or $E$ out of $5$ boxes.

Because balls are thrown uniformly at random, none of the $4$ special boxes are more or less likely to be empty than the other. Thus, we can use linearity of expectation to find the expected number of empty boxes using the probability we found before.

Answer: $4\times\frac35^{10}$