# Lottery Winning Numbers

There is a generic formula for determining the number is permutations (and thus combinations) of x choices from a set of y items, as follows:

x=? Permutations --- --------------------------------------- x=1 y x=2 y^2 -y x=3 y^3 -3y^2 +2y x=4 y^4 -6y^3 +11y^2 -6y x=5 y^5 -10y^4 +35y^3 -50y^2 +24y x=6 y^6 -15y^5 +85y^4 -225y^3 +274y^2 -120y For example, in a 6/49 lottery, your chances of winning the jackpot (i.e. of correctly selecting the winning combination of 6 numbers from a choice of 49) is just under 1 in 14 million, derived as follows: 49^6 = 13,841,287,201 -15*49^5 = -4,237,128,735 + 85*49^4 = 490,008,085 + -225*49^3 = -26,471,025 + 274*49^2 = 657,874 + -120*49 = -5,880 + -------------- 10,068,347,520 = 720 / (N.B. equivalent to 6!) ============== 13,983,816 =

Note 1: your lottery mileage may vary. The odds of winning your favorite lottery just might be better than in many examples, but given the trends in lottery design your odds may well be worse. Possibly much worse. For instance, "Powerball" is a very popular multi-state lottery in the United States, which is known for jackpots that grow obscenely large from time to time. This attractive feature is made possible simply by designing the game to be diabolically difficult to win: 1 chance in 120,526,770. That's almost nine times worse than winning odds in the most common 6/49 lottery version. It is the result of a design that is somewhat more complex.

Powerball lottery players also pick six numbers, but two different "bags" are used. The first five numbers come from one bag that contains numbers from 1 to 53 (the order doesn't matter). The sixth number - the "Powerball number" - comes from the second bag, which contains numbers from 1 to 42. When the drawing is held, your first five lottery numbers must match the first five drawn and your Powerball number must match the Powerball number drawn, or else you fail to win the jackpot. In other words, it is not good enough to pick 10, 18, 25, 33, 42, 7 when the drawing is 7, 10, 25, 33, 42, 18. Even though you had all the right lottery numbers, the Powerball number at the end of your ticket doesn't match the one drawn, so you would be credited with matching only four numbers (10, 25, 33, 42). More about Powerball winning numbers.

Note 2: the jackpot is usually not the only prize. Most lotteries give lesser prizes for matching just some of the winning numbers. The Powerball game described above is an extreme case, giving a very small payout (US\$3) even if you match only the Powerball number at the end of your ticket. The odds of doing this are refreshing: 1 in 70. Match more numbers and the payout goes up. Although none of these additional prizes affect the chances of winning the jackpot, they do improve the odds of winning something and therefore they add a little to the value of the ticket.