Question

In a certain test there are $n$ questions. In this test $2^k$ students gave wrong answer to at least $(n-k)$ questions, where $k=01,2,..n$. If the total number of wrong answers is $4095,$ the value of $n$ is

• A. $11$
• B. $12$
• C. $13$
• D. $15$

Answer 1

Answer: (B)

$12$

The number of students answering at least r questions incorrectly is $2^{n-r}$

Thus the number of students answering exactly r $(0\le r\le n-1)$ question incorrectly is $2^{n-r}-2^{n-(r+1)}$.

Also, the total number of wrong answers is $=1(2^{n-1}-2^{n-2})+2(2^{n-2}-2^{n-3})+3(2^{n-3}-2^{n-4})+....+(n-1)(2^1-2^0)+n{2}^0$

$=2^{n-1}+2^{n-2}+2^{n-3}+....+2^0=2^n-1$

$\Rightarrow 2^n-1=4095\Rightarrow 2^n=4096\Rightarrow n=12$