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In a certain test, there are n questions. In this test, `2^(n-k)` students gave wrong answers to atleast k questions, where k=1,2,3, . . ,. If the total number of wrong answers given in 127, then the value of n is

Text Solution

Verified by Experts

The correct Answer is:
7
The number of student answering exactly `i(1 le I le n-1)`
questions wrongly is `2^(n-1)-2^(n-i-1)`. The number of students answering all n questions wrongly is `2^(0)`.
Hence, the total number of wrong answer.
`underset(i=1)overset(n-1)(sum)i(2^(n-1)-2^(n-i-1)+n(2^(0)))=127`
`implies2^(n-1)+2^(n-2)+2^(n-3)+ . . .+2^(1)+2^(0)=127`
`implies 2^(n)-1=127`
`implies2^(n)=128=2^(7)`
`therefore n=7`
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