A store has an introductory sale on 12 types of candy bars. A customer may choose one bar of any five different types and will be charged no more than $1.75. Show that although different choices may cost different amounts, there must be at least two different ways to choose so that the cost will be the same for both choices.
2026-04-01 08:01:35.1775030495
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You find the number of possible choices (the number of "pigeons") and the number of possible prices (the number of "pigeonholes"), and see which one is greater.
There are $_{12}C_5 = 792$ choices (choose 5 from a set of 12 with order not mattering). The number of possible total prices is at most 176, in one-cent increments from \$0.00 up to \$1.75. So no assignment of prices to the items can yield unique total prices for every selection of choices, by the Pigeonhole Principle.