Scheduling the Best Episode

A television series has $n+1$ episodes, where $n\geqslant 10$. After watching the first episode, a viewer plans the rest as follows:

1. counting days from the day after finishing the first episode, the remaining $n$ episodes are distributed at random over $2n$ days; 2. on each day the viewer watches either nothing or exactly one complete episode; 3. at most one episode is watched per day.

The most exciting part of the series is episode $n$. Let $X$ be the day (counted as above) on which episode $n$ is watched.

Part 1. Give the distribution of $X$, and prove that the most likely day for watching episode $n$ is day $2n-2$.

Part 2. Find $E(X)$.