In a certain test there are n questions. In this test $2^k$ students gave wrong answers to at least$(n – k)$ questions, where$k = 0, 1, 2, \dots , n$. If the total number of wrong answers is $4095$, then value of$n$is

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

$\therefore$ the number of students answering exactly $r(1\le r\le n-1)$ questions incorrectly is  $2^{n-r}-2^{n-(r+1)}$

Also, the number of students answering all questions wrongly is $2^0=1$

Thus, the total number of wrong answers is

$\begin{array}{l}1\left(2^{n-1}-2^{n-2}\right)+2\left(2^{n-2}-2^{n-3}\right)+3\left(2^{n-3}-2^{n-4}\right)+\dots .....+(n-1)\left(2^1-2^0\right)+n\left(2^0\right)\\ \\ =2^{n-1}+2^{n-2}+\dots +2^0=2^n-1\end{array}$

Now, $2^n-1=4095\Rightarrow 2^n=4096=2^{12}\Rightarrow n=12$