Question

In a certain test there are n questions.In this test $2^{n-1}$ students gave the wrong answers to at least i questions, where $i=1,2,\mathrm{3.....}n$. If the total number of wrong answer given is $2047$ then n is equal to

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

Answer 1

The correct option is B $11$

The number of student answering exaclty $i(l\le i\le n-1)$ question wrongly is $2^{n-1}{-}^{n-i-1}$.
The number of student answering all n question wrongly is $2^{\circ }$
$\therefore$ total number of wrong answer is
$1(2^{n-1}-2^{n-1})+2(2^{n-2}-2^{n-3})+...+(n-1)(2^1-2^{\circ })+n\cdot 2^{\circ }$
$=2^{n-1}+2^{n-2}+...+2+1$
$\therefore 2047=2^n-1$
$\Rightarrow n=11$