Question

In a test, there are $n$ questions in which $2^{n-i}$ students gave wrong answers to atleast $i$ questions, where $i=1,2,....n.$ If the total number of wrong answers given is $2047$, then $n$ is equal to

• A. $12$
• B. $11$
• C. $10$
• D. None of these

Answer 1

The correct option is B $11$

The number of students giving exactly i wrong answers $=2^{n-i}-2^{n-i-1}$
Hence, total number of wrong answers
$=\sum _{i=1}^{n}i\times (2^{n-i}-2^{n-i-1})+n\times 1$
Therefore, $\sum ^{n}i=1i\times (2^{n-i}-2^{n-i-1})+n=2047$
$\Rightarrow 1(2^{n-1}-2^{n-2}+2(2^{n-2}-2^{n-3})+.....+(n-1\left)\right(2-1)+n=2047$
$\Rightarrow 2^{n-1}+2^{n-2}+....+2-(n-1)+n=2047$
$\Rightarrow 1+2+2^2+....+2^{n-1}=2047$
$\Rightarrow \frac{2^n-1}{2-1}=2047$
$[\because \text{sum of GP series},S_n=a\frac{(r^n-1)}{r-1},\text{when}r>1]$
$\Rightarrow 2^n=2048$
$\Rightarrow 2^n=2^{11}$
$\Rightarrow n=11$