Q: In a certain test, there are n questions. In this test 2^{n-i} students gave wrong answers to atleast i question, 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

Sol. The number of students answering exactly i ( 1≤ i ≤ n-1) questions wrongly is 2^{n-i} – 2^{n-i-1}. The number of students answering all n questions wrongly is 2^{0} . Thus, the total number of wrong answers is

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 = 2047

2^{n} = 2048 = 2^{11}

n = 11

Hence (B) is the correct answer.