A monthly state lottery picks five distinct integers from $1$ to $30$ and selects one of the five

to be the bonus number. An entry consists of five distinct integers from $1$ to $30$ with one

of the numbers designated as the bonus number.

Each month there are $100,000$ entries. Each entry that matches the five distinct numbers is

awarded $50,000.$ If the bonus number is also matched, an additional $250,000$ is awarded

to that entry. Nothing is awarded for matching fewer than five numbers.

Calculate the expected payout from the lottery in a month.