JohnnyLang™

Johnny is writing his own programming language: JohnnyLang™. He starts thinking about hash tables . He decides to implement a hash table that resolves collisions by chaining. For the simplicity, he decides that it will implement simple uniform hashing ( SUHA ) i.e., each key is equally likely to be hashed into each slot. Suppose we have a table of size $2n$ and $n$ keys are inserted into the table. The expected number of elements that hash into any slot $i$ can be expressed as a fraction $\dfrac{a}{b}$ , where $a$ and $b$ are coprime positive integers . Find $a+b$ .