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Question: In a centre test, there are \(P\) questions. In this, \({2^{P - r}}\) students give wrong answers to...

In a centre test, there are PP questions. In this, 2P−r{2^{P - r}} students give wrong answers to at least rr questions (1⩽r⩽P)(1 \leqslant r \leqslant P). If total number of wrong answers given is 20472047, then the value of PP is
A) 1414
B) 1313
C) 1212
D) 1111

Explanation

Solution

We are given the number of students who give wrong answers to at least rr question, so to find the exact wrong questions, subtract the number of students having r−1r - 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 = PP
2P−r{2^{P - r}}students give wrong answers to at least rr questions.
Which means
At r=1r = 1
2P−1{2^{P - 1}} students have at least one question wrong.
At r=2r = 2
2P−2{2^{P - 2}} students have at least 22 questions wrong.
At r=3r = 3
2P−3{2^{P - 3}} students have at least 33 questions wrong.
And so on---
At r=Pr = P
2P−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−1r - 1 wrong questions from the number of students having rr wrong questions
2P−1−2P−2{2^{P - 1}} - {2^{P - 2}}students got exactly 11 question wrong.
2P−2−2P−3{2^{P - 2}} - {2^{P - 3}} students got exactly 22 questions wrong.
Similarly,
2P−3−2P−4{2^{P - 3}} - {2^{P - 4}} students got exactly 33 questions wrong.
And so on--
2P−(P−1)−2P−P{2^{P - (P - 1)}} - {2^{P - P}} students got exactly P−1P - 1 questions wrong.
So, total number of wrong answers are:
1(2P−1−2P−2)+2(2P−2−2P−3)+3(2P−3−2P−4)+−−−−+(P−1)(2P−(P−1)−2P−P)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
2P−1+2P−2+2P−3+−−−−+2P−P{2^{P - 1}} + {2^{P - 2}} + {2^{P - 3}} + - - - - + {2^{P - P}}
And the sum of the number of wrong questions is given 20472047 in the question statement.
2P−1+2P−2+2P−3+−−−−+2P−P=2047{2^{P - 1}} + {2^{P - 2}} + {2^{P - 3}} + - - - - + {2^{P - P}} = 2047
This is nothing but
2P−1=2047 2P=2048 P=11  {2^P} - 1 = 2047 \\\ {2^P} = 2048 \\\ P = 11 \\\
therefore, the value of P=11P = 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.