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(B) Let $S_i$ be the number of students who answered at least $i$ questions wrongly. We are given $S_i = 2^{n-i}$ for $i = 1, 2, \dots, n$.The number

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$11$

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(B) Let $S_i$ be the number of students who answered at least $i$ questions wrongly. We are given $S_i = 2^{n-i}$ for $i = 1, 2, \dots, n$.The number of students who answered exactly $i$ questions wrongly is $N_i = S_i - S_{i+1}$ for $1 \le i The total number of wrong answers is given by the sum $\sum_{i=1}^{n} i \cdot N_i$.Total wrong answers $= 1(S_1 - S_2) + 2(S_2 - S_3) + 3(S_3 - S_4) + \dots + (n-1)(S_{n-1} - S_n) + n(S_n)$.Expanding this sum: $S_1 + S_2 + S_3 + \dots + S_n$.Substituting $S_i = 2^{n-i}$:Total wrong answers $= 2^{n-1} + 2^{n-2} + \dots + 2^0$.This is a geometric series with sum $\frac{2^n - 1}{2 - 1} = 2^n - 1$.Given $2^n - 1 = 2047$,we have $2^n = 2048 = 2^{11}$.Therefore,$n = 11$.

In a certain test,there are $n$ questions. In the test,$2^{n-i}$ students gave wrong answers to at least $i$ questions,where $i = 1, 2, \dots, n$. If the total number of wrong answers given is $2047$,then $n$ is equal to

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