Total number of wrong answers permutation

In summary, the total number of wrong answers given in a test with n questions is 2047, with 2n-1 students giving wrong answers to at least 1 question and 2n-2 students giving wrong answers to at least 2 questions. Using the equation 2047 = 2^(n+1) - 1, it can be determined that n = 11, and one student got all 11 questions wrong. This pattern can be extended to find the number of students who got at least i questions wrong, where i = 1, 2, 3, ..., n.
  • #1
zorro
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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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

1. What does "total number of wrong answers permutation" mean?

The total number of wrong answers permutation refers to the number of ways that a given set of incorrect answers can be arranged within a multiple-choice test or quiz.

2. How is the total number of wrong answers permutation calculated?

The total number of wrong answers permutation is calculated using the formula nPr = n! / (n-r)!, where n represents the total number of possible answers and r represents the number of incorrect answers.

3. Why is the total number of wrong answers permutation important?

The total number of wrong answers permutation is important because it helps determine the likelihood of guessing the correct answer on a multiple-choice test. It can also provide insights into how well a student understands the material if they consistently choose the same incorrect answer.

4. Can the total number of wrong answers permutation be used to improve test scores?

Yes, the total number of wrong answers permutation can be used to improve test scores by identifying patterns in incorrect answers and providing guidance on how to approach multiple-choice questions more strategically.

5. Are there any limitations to using the total number of wrong answers permutation?

Yes, there are limitations to using the total number of wrong answers permutation. This calculation assumes that all answers are equally likely to be chosen and does not take into account any other factors that may influence a student's performance on a test, such as test-taking strategies or prior knowledge.

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