Question

In a certain test, there are $n$ questions. In the test $2^{n-i}$ students gave wrong answers to at least $i$ questions, where $i=1,2,...,n$. 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)

$11$

We have the following information:
$2^{n-1}$ students got at least $1$ question wrong
$2^{n-2}$ students got at least $2$ questions wrong
$2^{n-3}$ students got at least $3$ questions wrong
...
$2^{n-n}$ students got at least $n$ questions wrong i.e. all $n$ questions wrong
Thus,
$2^{n-1}-2^{n-2}$ students got exactly $1$ question wrong
$2^{n-2}-2^{n-3}$ students got exactly $2$ questions wrong
$2^{n-3}-2^{n-4}$ students got exactly $3$ questions wrong
...
$2^{n-n+1}-2^{n-n}=2^1-2^0$ students got exactly $n-1$ questions wrong
$2^{n-n}=2^0$ students got exactly $n$ questions wrong.
Thus, total number of wrong answers:
$1\ast (2^{n-1}-2^{n-2})+2\ast (2^{n-2}-2^{n-3})+3\ast (2^{n-3}-2^{n-4})+...+(n-1)\ast (2^1-2^0)+(n\ast 2^0)=2047$
$=>2^{n-1}+2^{n-2}+...+2^0=2047$
$=>2^n-1=2047=>2^n=2048=>n=11$
Hence, (b) is correct.