Question

In a centre test, there are $P$ questions. In this, ${2^{P - r}}$ students give wrong answers to at least $r$ questions $(1 \leqslant r \leqslant P)$. If total number of wrong answers given is $2047$, then the value of $P$ is
A) $14$
B) $13$
C) $12$
D) $11$

Answer 1

Hint: We are given the number of students who give wrong answers to at least $r$ question, so to find the exact wrong questions, subtract the number of students having $r - 1$ wrong questions from the number of students having r wrong questions then add all the number of wrong answers and equate it to 2047.

Complete step by step answer:
We are given
Total number of questions in a centre test = $P$
${2^{P - r}}$students give wrong answers to at least $r$ questions.
Which means
At $r = 1$
${2^{P - 1}}$ students have at least one question wrong.
At $r = 2$
${2^{P - 2}}$ students have at least $2$ questions wrong.
At $r = 3$
${2^{P - 3}}$ students have at least $3$ questions wrong.
And so on---
At $r = P$
${2^{P - P}}$students have at least P questions wrong.
Now to find the exact questions wrong by the students, subtract the number of students having $r - 1$ wrong questions from the number of students having $r$ wrong questions
${2^{P - 1}} - {2^{P - 2}}$students got exactly $1$ question wrong.
${2^{P - 2}} - {2^{P - 3}}$ students got exactly $2$ questions wrong.
Similarly,
${2^{P - 3}} - {2^{P - 4}}$ students got exactly $3$ questions wrong.
And so on--
${2^{P - (P - 1)}} - {2^{P - P}}$ students got exactly $P - 1$ questions wrong.
So, total number of wrong answers are:
$1({2^{P - 1}} - {2^{P - 2}}) + 2({2^{P - 2}} - {2^{P - 3}}) + 3({2^{P - 3}} - {2^{P - 4}}) + - - - - + (P - 1)({2^{P - (P - 1)}} - {2^{P - P}})$
Which is
${2^{P - 1}} + {2^{P - 2}} + {2^{P - 3}} + - - - - + {2^{P - P}}$
And the sum of the number of wrong questions is given $2047$ in the question statement.
${2^{P - 1}} + {2^{P - 2}} + {2^{P - 3}} + - - - - + {2^{P - P}} = 2047$
This is nothing but
$
  {2^P} - 1 = 2047 \\
  {2^P} = 2048 \\
  P = 11 \\
 $
therefore, the value of $P = 11$
So the total number of questions in a centre test are 11.
And hence option D is correct.


Note:
A permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.