Question

In a certain test, there are total n questions. In the test ${{\text{2}}^{{\text{n - i}}}}$ students gave wrong answers to at least i questions where i=1,2....n. If the total number of wrong (incorrect) answers given is 2047, then n is equal to
A. 10
B. 11
C. 12
D. 13

Answer 1

Hint-For every value of 'i' we need to find the number of students in power of 2 then add all the cases to make it 2047. Then apply formula for the sum of the cases of geometric progression. Then find the value of n which is asked.

Complete step by step solution:
In the question, we are told that in a certain test, there are total n questions. Now, we are informed that for at least 'i' questions, where i contain all the values from ‘1’ to ‘n’, total ${2^{n - i}}$. Student gave wrong answers. Now we are given a total number of wrong answers given is 2047, so we have to find a value of n.
So according to the given question for at least 1 question ${2^{n - 1}}$ students gave wrong answer, for at least 2 questions ${2^{n - 2}}$ students gave wrong answer similarly, for at least 3 questions the number of students will be ${2^{n - 3}}$ ways and so on, and finally for at least n questions the number of students will be ${2^{n - n}}or{2^0}$ or 1 way.
Now,
${2^0} = {2^1} - 1$
${2^0} + {2^1} = {2^2} - 1$
${2^0} + {2^1} + {2^2} = {2^3} - 1$
Or the sum of powers of 2 equals 1 less than the next power,
∵2048=${2^{11}}$ and 2047 is one less than ${2^{11}}$
 ∴ it is the sum of powers of 2 till n=11.
Hence, the answer is 11.

Note- In this question, students should be careful while considering all the cases which add up to 2047 as missing out any case can result in a change of value n. and also avoid basic calculations mistakes during calculations.