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PrintThe 4th Japanese Junior Mathematical Olympiad
Japan counting and probability
Problem
There are ten red cards, numbered , , , , and ten blue cards, also numbered , , , . How many ways are there to choose three from these twenty cards so that the sum of the numbers on the cards chosen is or less?
Solution
Write , , , or on the back of the cards so that each card has two numbers which add up to .
Call a set of three cards good if the sum of the numbers on their faces is or less, and call it bad if the sum of the numbers on their backs is or less.
Since the sum of their faces and their backs add up to , every set of three cards is good or bad, and none is both. Furthermore, by symmetry, there must be the same number of good and bad sets. Therefore, there are exactly ways of required choice.
Call a set of three cards good if the sum of the numbers on their faces is or less, and call it bad if the sum of the numbers on their backs is or less.
Since the sum of their faces and their backs add up to , every set of three cards is good or bad, and none is both. Furthermore, by symmetry, there must be the same number of good and bad sets. Therefore, there are exactly ways of required choice.
Final answer
570
Techniques
Counting two waysEnumeration with symmetry