The question states: In a lottery $5$ numbers are chosen from the set $\{1,..,90\}$ without replacement. We order the $5$ numbers in increasing order and we denote by $X$ the number of times the difference between two neighboring numbers is $1$.
I included a picture of my attempt and I think I might've messed up since the numbers are ordered.
Order cannot matter: it affects the numerator and denominator by the same factor of $5!$
So an expectation of $(5 \times 4)\times \dfrac{89}{90\times 89} = \dfrac29$, which is what you found, looks correct