Question

In a certain test there are n questions. In the test 2^n-1 students gave wrong answers to at least i questions, where i = 1, 2, ........n. If the total number f wrong answers given is 2047, then n is equal to

• A. 10
• B. 11
• C. 12
• D. 13

Answer 1

Answer: (B)

11

Answer 2

Since the number of student giving wrong answers to at least i question (i = 1,2, ........., n) = 2^n-i.

The number of students answering exactly i (1 ≤ i ≤ - 1) questions wrongly = {the number of students answering at least i questions wrongly, i = 1, 2, ........., n)} – {the number of students answering at least (i + 1) questions wrongly (2 ≤i + 1 ≤n)} = 2^n-i – 2^n-(i+1) (1 ≤ i ≤ n – 1).

Now, the number of students answering all the n questions wrongly = 2^n-n = 2⁰.

Thus the total number of wrong answers

= 1(2^n-1 – 2^n-2 + 2(2^n-2 – 2^n-3) + 3(2^n-3 – 2^n-4) + ....... + (n – 1) (2¹ – 2⁰) + n(2⁰)

= 2^n-1 + 2^n-2 + 2^n-3 + ........+ 2⁰ = 2ⁿ –1 ($\because$Its a G.P.)

∴ As given 2ⁿ –1 = 2047 ⇒2ⁿ = 2048 = 2¹¹ ⇒ n =11.