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 $
$\Rightarrow n = 11$