EXAMPLE 17 Winning the Texas Lottery?

Most states have lottery games. The Texas Lottery (www.txlottery.org) has a Lotto Texas game that involves choosing six numbers from the set of whole numbers from 1 to 54. You win (at least a share of) the jackpot so long as the collection of numbers you pick is the same collection that the lottery selects. Repetition is not allowed—that is, the same number can’t be picked twice in the same drawing. Unlike permutations, order does not matter here. A person who chooses {49, 12, 18, 26, 22, 5} will receive the same payout as a person who chooses {26, 18, 12, 49, 5, 22}.

This is a perfect setup for a combination because the order of the selected numbers doesn’t matter. The number of collections of lottery balls is “54 choose 6,” or

possible sets of numbers. Because only one of these sets of numbers will correspond to the jackpot, the probability of your ticket winning (at least a share of) the jackpot is .

More generally, the scenario of choosing an unordered subset of balls from a collection of different balls is a combination.