The national and state lotteries, along with Powerball, use derangements in every single game that they play. Derangements are perfect for them because they need to be able to find out the number of possibilities where absolutely nothing will match at all.
       Let's say that the lottery is devising a new game. This game has an array of 7x7 scratch squares with different colors under them, and costs $1.50 to play. The rule that the lottery makes for this game is that you must only scratch one color in each column. You can not duplicate colors as you are scratching across the rows. The restrictions that they set on this game are that if you get one color in the row right, you receive $0.10. If you get two correct, you get four times as much, and so on. That way, if you pick all seven squares correctly, you will receive $ 307.20. The payout here isn't that great because the chances of winning are easier, or that is what we tend to believe.
       For the very fist game, the lottery picks the colors in sequential order, ROYGBIV. They can easily calculate what the odds of you winning are. First they see that the possible combinations of picking colors are 7! = 5040. Next they will find out how many ways that no one will win. This is from the derangement of !7 = 1854. This means that out of the possible 5040 combinations, there are 1854 ways that you will choose seven colors, and have none of them match the original combination. Now, the only way that the lottery company can lose money is if someone gets three or more colors correct. The next thing the lottery will do is see how many ways to get exactly one correct. This is by finding !6 which equals 265. Then multiplying that by seven people - choose one: 7*265 = 1855 ways that exactly one color will match. For exactly two matches we have !5 which equals 44, multiplied by seven people - choose two: 21*44 = 924 ways that exactly two correct.
       Adding these up, we find that 1845 + 1855 + 924 = 4624. 4624/5040 = .917. Now the lottery will use this to show that this new game has a 91.7% chance that you will guess at most only two colors. So on each game that they sell, they have about a 9/10 chance of winning.
Lotteries and the Powerball will use their odds often to calculate how tough their games are. Most of the games that you play will have absolutely horrible odds like this. Most people won't sit down and calculate the odds, they will just look at how much they could possibly win.
       The real world application here shows that lotteries along with the Powerball will caclulate often in new games the number of ways where no one will pick anything correctly.