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, …, 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 ≤ r ≤ n – 1) questions incorrectly is 2^n–r – 2^{n–(r + 1)}.
Also, the number of students answering all questions wrongly is 2⁰ = 1.
Thus, the total number of wrong answers is
$1 (2^{n - 1} - 2^{n - 2}) + 2 (2^{n - 2} - 2^{n - 3}) + 3 (2^{n - 3} - 2^{n - 4}) + \dots + (n - 1) (2^{1} - 2^{0}) + n (2^{0}) = 2^{n - 1} + 2^{n - 2} + \dots + 2^{0} = 2^{n} - 1 .$
Now, 2ⁿ – 1 = 4095
$\begin{array}{l} \Rightarrow 2^{n} = 4096 = 2^{12} \\ \Rightarrow n = 12 . \end{array}$

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