Elaboration:
With both our formulae I get the 16 Billion figure.
The only difference is in the division of your answer by five, then by four, then by three and then by two. This is because the chance of your first number coming out is not 1 in 52, it’s 5 in 52 (you have 5 chances), then 4 in 51, 3 in 50, 2 in 19 and 1 in 48. You then have a 1 in 52 chance of hitting the mega ball.
Can you see the division line with 543*2 under it in my post?
I meant 2 in 49 instead of 2 in 19 in that post - sorry for any confusion.
That assumes that every dollar taken in ticket sales goes into the jackpot, which can’t be true, as I mention below.
The reason that lotteries are “a tax for those who can’t do math” is not that you can’t predict the winning number, but that the expected value of winnings is lower than the cost of a ticket. You said yourself that the house always wins. That is, if you bought every lottery ticket, you would the sole winner, but the winnings would be less than the amount you spent on the tickets, because the state keeps some as revenue. (That is a little different than this thread’s OP, which asks about buying every possible combination and assuming that there are a lot of other losers out there to fund the jackpot.)
This fails to take into account that the jackpot is built up over a number of draws where it is not won. Theoretically, the cost of hitting all combinations after several rollover jackpots may be less than the jackpot. While doing this just isn’t feasible in virtually all modern lotteries, it’s not true to say that it’s mathematically built into the system that the state wins on every single draw.