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

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 2⁰ . Thus, the total number of wrong answers is

1(2^n-1 – 2^n-2) + 2(2^n-2 – 2^n-3 ) + …. + (n-1)(2¹ – 2⁰) + n (2⁰)

= 2^n-1 + 2^n-2 + 2^n-3 + …..+ 2⁰

= 2ⁿ – 1

Thus , 2ⁿ – 1 = 2047

2ⁿ = 2048 = 2¹¹

n = 11

Hence (B) is the correct answer.