Question

In a certain test there are $\mathrm{n}$ questions. In this test $2^{\mathrm{k}}$ students gave wrong answers to at least $\mathrm{n}–\mathrm{k}$ question.$\mathrm{K}=0,1,2...\mathrm{n}$. If the number of wrong answers is $4095$ then value of $\mathrm{n}$ is?

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

Answer 1

The correct option is B

$12$

Explanation for correct option:

Finding the value of $\mathbf{n}$:

The number of students who answered at least $\mathrm{r}$ questions incorrectly is $2^{\mathrm{n}-\mathrm{r}}$.

Now, the number of students who answered exactly $\mathrm{r}(0\le \mathrm{r}\le \mathrm{n}-1)$ questions incorrectly is $2^{\mathrm{n}-\mathrm{r}}–2^{\mathrm{n}-(\mathrm{r}+1)}$

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

$$\begin{gathered} 2^{\mathrm{n}}-1=4095 \\ \Rightarrow 2^{\mathrm{n}}=4096 \\ \Rightarrow 2^{\mathrm{n}}=2^{12} \\ \Rightarrow \mathrm{n}=12 \end{gathered}$$

Hence, the correct option is B.