Question

In a certain test, there are $n$ questions. In this test $2^{n-i}$ students gave wrong answers to at least $i$ questions; where $i=1,\mathrm{2.............}n-1,n$. If the total number of wrong answers given is $2047$, then $n$ is equal to

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

Answer: (B)

The no. of students answering exactly $i\left(1\le i\le n-1\right)$ questions wrongly is $2^{n-i}-2^{n-i-1}$. The no. of students answering all n questions wrongly is $2^{\circ }$. Thus, the total number of wrong answer is
$1\left(2^{n-1}-2^{n-2}\right)+2\left(2^{n-2}-2^{n-3}\right)+\dots \left(n-1\right)\left(2^1-2^{\circ }\right)+n\left(2^{\circ }\right)$
$\Rightarrow 2^{n-1}+2^{n-2}+2^{n-3}+.........+2+1=2^n-1$
Thus $2^n-1=2047$
$\Rightarrow 2^n=2048=2^{11}$
$\therefore n=11$