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(B) Let $S_i$ be the number of students who gave wrong answers to at least $i$ questions. We are given $S_i = 2^{n-i}$.The total number of wrong answe

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$11$

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(B) Let $S_i$ be the number of students who gave wrong answers to at least $i$ questions. We are given $S_i = 2^{n-i}$.The total number of wrong answers is the sum of the number of students who gave wrong answers to at least $i$ questions for all $i$ from $1$ to $n$.Total wrong answers $= \sum_{i=1}^{n} S_i = \sum_{i=1}^{n} 2^{n-i}$.This is a geometric series: $2^{n-1} + 2^{n-2} + \ldots + 2^0$.Using the sum formula for a geometric series $\sum_{k=0}^{n-1} 2^k = \frac{2^n - 1}{2 - 1} = 2^n - 1$.Given that the total number of wrong answers is $2047$,we have $2^n - 1 = 2047$.$2^n = 2048$.Since $2048 = 2^{11}$,we get $n = 11$.

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

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