Total number of wrong answers permutation

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The problem involves determining the number of questions, n, in a test where 2n – i students answered at least i questions incorrectly, with a total of 2047 wrong answers given. The key insight is recognizing that 2047 can be expressed as 2^11 - 1, which relates to the powers of 2 in the context of the problem. By analyzing the distribution of wrong answers, it is concluded that n equals 11, as this corresponds to the scenario where one student answered all questions incorrectly. The discussion emphasizes the importance of rephrasing the problem and identifying patterns in the data. Ultimately, the solution hinges on understanding the relationship between the number of students and the questions answered incorrectly.
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Homework Statement


In a test there were n questions. In the test 2n – i students gave wrong answers to at least i questions where i = 1, 2, 3, …, n. If the total number of wrong answers given is 2047, then n is


The Attempt at a Solution



I don't understand how to transform the second sentence to an equation.
 
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Instead of transforming the 2nd sentence into an equation, what about rephrasing the 2nd sentence in terms of specific values of i? For example, what does this sentence say when i = 1? i = 2? ... i = n?

It may or may not be relevant that 2047 = 211 - 1 = 20 + 21 + 22 + ... + 210.
 
2n-1 students gave wrong answers to atleast 1 question.
2n-2 students gave wrong answers to atleast 2 questions...

Do I have to add 2n-1 + 2n-2 + ... 20?
It may or may not be relevant that 2047 = 211 - 1 = 20 + 21 + 22 + ... + 210.

Yes 11 is the answer.
 
Going from the other end, 2n - n = 20 = 1 student got at least n questions wrong, which means that 1 of the n students got all n questions wrong. In this case, "at least n" means "exactly n" since there were only n questions.

Work out some more sentences and see if you can find a pattern.
 

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