An airline sells $120$ tickets for a flight that seats $100.$ Each ticket is non$-$refundable and costs $$250.$ The unit cost of flying a passenger $($fuel$,$ landing fees, etc.$)$ is $$110$ and assumes that every flight is full. If the flight is overbooked, each person who does not get a seat is given $$400$ in cash. Assume it is equally likely that any number of people between $91$ and $120$ show up for the flight. Rounded to the nearest thousand, what is the average expected profit the flight will generate?

$$10,000$

$$14,000$

$$16,000$

$$22,000$