• 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

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$