Competition problem — Combinatorics

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Combinatorics Difficulty 3.9 AMC 10/12 Find the answer Italy

Problem:

One and only one of the following statements is false. Which one?

(A) "(B) is false"
(B) "(C) is false"
(C) "(E) is true"
(D) "(A) is true"
(E) "Three of the preceding are true".

This was a multiple-choice question, but the options didn't survive into the source we have. The answer given is B, and the solution below works it through.

Solution

Solution:

The answer is (B). If we suppose that the statement "(B) is false" is false, then (D) is also false, against the hypothesis of the problem. Therefore the statement "(B) is false" is true, hence the false statement is (B) and all the others turn out to be true.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.