Question

In a test there were $n$ questions. In the test $2^{n-i}$ students gave wrong answers to at least $i$ questions $i=1,2,3....n$. If the total number of wrong answers given is $2047$, then $n$ is

• A. $12$
• B. $11$
• C. $10$
• D. $13$

Answer 1

Answer: (B)

$11$

Let the number of students who gave wrong answer to exactly $i$ questions be $E_i$

Number students who gave wrong answer to at least $1$ question $=2^{n-1}=E_1+E_2+E_3+....E_n$
Number students who gave wrong answer to at least $2$ questions $=2^{n-2}=E_2+E_3+E_4+...+E_n$
Number students who gave wrong answer to at least $3$ questions $=2^{n-3}=E_3+E_4+E_5+...E_n$
:
:
Number students who gave wrong answer to at least $n$ questions $=1=E_n$
Number of wrong answers given $=E_1+2E_2+3E_3+...+n{E}_n$
Adding $n$ equations listed above we have,

$E_1+2E_2+3E_3+...+n{E}_n=1+2+2^2+2^3+..{.2}^{n-1}$
Hence,
Sum$=1+2+2^2+2^3+..{.2}^{n-1}$
$2047=\frac{2^n-1}{2-1}$
$2047=2^n-1$
$\therefore 2^n=2048$
$\therefore n=11$