Question

In a certain examination paper, there are n questions. For j = 1,2 …n, there are \( 2^{n-j} \) students who answered j or more questions wrongly. If the total number of wrong answers is 4095, then the value of n is

• A. 12
• B. 11
• C. 10
• D. 9

Hint:

Think about how many times a student with k wrong answers is counted when you add up the groups "j or more wrong". The total then becomes a simple geometric series.

Answer 1

The Correct Option is A

Step 1: Understand what is being counted:
There are n questions and some number of students. The paper tells us how many students answered j or more questions wrongly, for each j from 1 to n. That number is \(2^{n-j}\).
We are told the total number of wrong answers (all students together) is 4095, and we must find n.

Step 2: Key idea: count the wrong answers in a different way:
Take a student who got exactly k questions wrong. That student gives k wrong answers in total.
The same student is also counted once in each of the groups "1 or more wrong", "2 or more wrong", ..., "k or more wrong". That is k groups.
So every wrong answer of every student is counted exactly once if we add up the group sizes for j = 1, 2, ..., n.

Step 3: Write the total as a sum:
Total wrong answers = (students with 1 or more wrong) + (students with 2 or more wrong) + ... + (students with n wrong).
\[ \text{Total} = 2^{n-1} + 2^{n-2} + \cdots + 2^{1} + 2^{0} \]

Step 4: Sum the geometric series:
The terms are 1, 2, 4, ..., \(2^{n-1}\). That is a geometric series with first term 1, common ratio 2 and n terms.
\[ \text{Total} = \frac{2^n - 1}{2 - 1} = 2^n - 1 \]

Step 5: Use the given total:
We are given that the total is 4095, so \(2^n - 1 = 4095\), which gives \(2^n = 4096\).
Now \(2^{10} = 1024\), \(2^{11} = 2048\) and \(2^{12} = 4096\). So \(n = 12\).

Step 6: Why the other options fail:
If \(n = 11\), the total is \(2^{11} - 1 = 2047\), which is only about half of 4095.
If \(n = 10\), the total is \(2^{10} - 1 = 1023\).
If \(n = 9\), the total is \(2^{9} - 1 = 511\).
Only \(n = 12\) gives 4095, which is option 1.

Final Answer:
The total number of wrong answers equals \(2^n - 1\). Setting it equal to 4095 gives \(n = 12\), which is option 1. \[ \boxed{n = 12 \; \text{(option 1)}} \]