Problem:
A game show offers a contestant three prizes , and , each of which is worth a whole number of dollars from to inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the order , , . As a hint, the digits of the three prices are given. On a particular day, the digits given were . Find the total number of possible guesses for all three prizes consistent with the hint.
Solution:
The number of possible orderings of the given seven digits is . These orderings correspond to seven-digit numbers, and the digits of each number can be subdivided to represent a unique combination of guesses for , , and . Thus, for a given ordering, the number of guesses it represents is the number of ways to subdivide the seven-digit number into three nonempty sequences, each with no more than four digits. These subdivisions have possible lengths , and their permutations. The first subdivision can be ordered in ways and the second and third in ways each, for a total of possible subdivisions. Thus the total number of guesses is or .
The problems on this page are the property of the MAA's American Mathematics Competitions