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 = 2 0 .

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) (2 1 – 2 0 ) + n(2 0 )

= 2 n-1 + 2 n-2 + 2 n-3 + ........+ 2 0 = 2 n –1 ( Its a G.P.)

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