Question

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

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

Answer 1

Answer: (B)

The number of students answering exactly
$i(1\le i\le n-1)$ questions wrongly is $2^{n-i}-2^{n-i-1}$.
The number of students answering all n questions wrongly is $2^0$.
$\therefore$ Total number of wrong answers is
$1(2^{n-1}-2^{n-2})+2(2^{n-2}-2^{n-3})+...$
$+(n-1)(2^1-2^0)+n\cdot 2^0$
$=2^{n-1}+2^{n-2}+...+2+1$
$\therefore 2047=2^n-1$
$\Rightarrow n=11$