First slide

Permutations

Question

In a certain test there are n questions. In this test $2^{k}$ students gave wrong answers to at least $(n - k)$ questions, where $k = 0, 1, 2, \dots, n$ . If the total number of wrong answers is $4095$ , then value of $n$ is

Moderate

A

11
B

12
C

13
D

15

Solution

The number of students answering at least $r$ questions incorrectly is $2^{n - r}$
$∴$ the number of students answering exactly $r (1 \leq r \leq n - 1)$ questions incorrectly is $2^{n - r} - 2^{n - (r + 1)}$
Also, the number of students answering all questions wrongly is $2^{0} = 1$
Thus, the total number of wrong answers is
$\begin{array}{l} 1 (2^{n - 1} - 2^{n - 2}) + 2 (2^{n - 2} - 2^{n - 3}) + 3 (2^{n - 3} - 2^{n - 4}) + \dots . . . . . + (n - 1) (2^{1} - 2^{0}) + n (2^{0}) \\ = 2^{n - 1} + 2^{n - 2} + \dots + 2^{0} = 2^{n} - 1 \end{array}$
Now, $2^{n} - 1 = 4095 \Rightarrow 2^{n} = 4096 = 2^{12} \Rightarrow n = 12$

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