JEE Main & Advanced Mathematics Permutations and Combinations Question Bank Critical Thinking Questions

  • question_answer 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

    Correct Answer: B

    Solution :

    Since the number of students giving wrong answers to at least \[i\] question\[(i=1,\ 2,........,n)={{2}^{n-i}}\]. The number of students answering exactly \[i\ (1\le i\le -1)\] questions wrongly = {the number of students answering at least \[i\] questions wrongly, \[i=1,\ 2,\ .........,n)\}\] - {the number of students answering at least \[(i+1)\] questions wrongly\[(2\le i+1\le n)\}\] \[={{2}^{n-i}}-{{2}^{n-(i+1)}}(1\le i\le n-1)\]. Now, the number of students answering all the \[n\] questions wrongly\[={{2}^{n-n}}={{2}^{0}}\]. Thus the total number of wrong answers \[=1({{2}^{n-1}}-{{2}^{n-2}}+2({{2}^{n-2}}-{{2}^{n-3}})+3({{2}^{n-3}}-{{2}^{n-4}})\]                                     \[+..........+(n-1)({{2}^{1}}-{{2}^{0}})+n({{2}^{0}})\] \[={{2}^{n-1}}+{{2}^{n-2}}+{{2}^{n-3}}+.........+{{2}^{0}}\]=\[{{2}^{n}}-1\]  \[(\because \ \text{Its}\ \text{a}\ \text{G}\text{.P}\text{.})\] \[\therefore \] As given \[{{2}^{n}}-1=2047\Rightarrow {{2}^{n}}=2048={{2}^{11}}\]\[\Rightarrow n=11\]


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