Total number of wrong answers permutation

[zorro]

Homework Statement

In a test there were n questions. In the test 2^{n – 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.

 

[Mark44]

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.

 

[zorro]

2^n-1 students gave wrong answers to atleast 1 question.
2^n-2 students gave wrong answers to atleast 2 questions...

Do I have to add 2^n-1 + 2^n-2 + ... 2⁰?

It may or may not be relevant that 2047 = 2¹¹ - 1 = 2⁰ + 2¹ + 2² + ... + 2¹⁰.

Yes 11 is the answer.

 

[Mark44]

Going from the other end, 2^{n - 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.