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  •  In a certain test there are questions. In this test <img alt="\mathrm{2^{k}}" src="https://entr

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

Option: 1

11


Option: 2

12


Option: 3

13


Option: 4

15


Answers (1)

best_answer

 The number of students answering at least r questions incorrectly is 2^{n-r}.

\therefore   The number of students answering exactly \mathrm{r(1 \leq r \leq n-1)}  questions incorrectly is \mathrm{2^{n-r}-2^{n-(r+1)}}.

Also, the number of students answering all questions wrongly is \mathrm{2^{0}= 1}.

Thus, the total number of wrong anser is

1\left(2^{n-1}-2^{n-2}\right)+2\left(2^{n-2}-2^{n-3}\right)+3\left(2^{n-3}-2^{n-4}\right)+\ldots+$ $(n-1)\left(2^{1}-2^{0}\right)+n\left(2^{0}\right)

=2^{n-1}+2^{n-2}+\ldots+2^{0}=2^{n}-1.

Now, 2^{n}-1=4095 \Rightarrow 2^{n}=4096=2^{12} \Rightarrow n=12.

Posted by

manish painkra

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