N a certain test, there are `n` question. In the test, `2^(n-i)` students gave wrong answers to at least `i` questions, where `i=1,,2ddot,ndot` if the total number of wring answers given is 2047, then `n` is equal to a. `10` b. `11` c. `12` d. `13`

Options

A. 10
B. 11
C. 12
D. 13

Explanation

The number of studnets answering exactly `k(1leklen-1)` questions wrongly is `2^(n-k)-2^(n-k-1)`. The number of studnets answering all questions wrongly is `2^(0)`. Thus, the total number of wrong answers
`=1.(2^(n-1)-2^(n-2))+2.(2^(n-2)-2^(n-3))+......+(n-1)(2^(1)-2^(0))+n.2^(0)`
`=2^(n-1)+2^(n-2)+2^(n-3)+......+2^(0)=2^(n)-`1
Thus, `2^(n)-1=2047implies2^(n)=2048implies2^(n)=2^(11)impliesn=11`