Problem 60 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In an examination a candidate has to pass in each of the papers to be successful. If the total number of ways to fail is 63 , how many papers are there in the examination? (A) 6 (B) 8 (C) 14 (D) None of these

Step-by-Step Solution

Answer

There are 6 papers in the examination.

Step 1: Understanding the Problem

The problem requires us to find the number of examination papers given that there are 63 ways for a candidate to fail. A candidate fails if they fail in at least one paper.

Step 2: Logical Analysis of Failing

If there are 'n' papers, for each paper the candidate has two outcomes: either pass or fail. Thus for 'n' papers, there are \( 2^n \) possible ways for the candidate's results. Failing in each paper is one of these, so the number of ways to fail in at least one paper is \( 2^n - 1 \).

Step 3: Setting the Equation

We are given that the number of ways to fail is 63. Thus, we have the equation \( 2^n - 1 = 63 \).

Step 4: Solving the Equation

We adjust the equation as follows: \( 2^n - 1 = 63 \) so \( 2^n = 64 \). Solve this equation by recognizing that \( 64 = 2^6 \). Thus, \( n = 6 \).

Step 5: Verification

The solution satisfies that \( 2^6 = 64 \). Therefore, \( 2^n - 1 = 63 \) holds true. This verifies our solution.

Key Concepts

Exponential Growth

Exponential growth is a powerful concept in mathematics where quantities increase rapidly due to continuous compounding. In the context of combinatorics and the original exercise, exponential growth is reflected in the number of possible outcomes when taking exams. For instance, when a student has a choice between passing or failing each paper, the number of different possible outcomes grows exponentially with each additional exam paper.

To visualize this, consider that with one paper, there are 2 possible outcomes (pass or fail). With two papers, this number doubles to 4 (pass-pass, pass-fail, fail-pass, or fail-fail). As we add more papers, the count follows the pattern of powers of two, namely, for 'n' papers, there are \(2^n\) possible outcomes.

This exponential growth significantly influences how we approach problems like the one in the exercise, making it clear that even a small increase in the number of papers results in a substantial increase in possible results.

Logical Reasoning

Logical reasoning is the process of using rational and systematic steps based on sound mathematical principles to come to a conclusion. In the problem at hand, logical reasoning helps break down the problem, allowing us to understand why the equation \(2^n - 1 = 63\) is used.

Let's delve into what this equation implies. For every paper in the exam, two possibilities exist: the student can either pass or fail. However, failing in any paper means failing the exam.

• This leads us to the understanding that the total number of possible outcomes where the student can fail at least one paper is \(2^n - 1\).
• The reason we subtract 1 is to exclude the scenario where the student passes all papers, that being only one outcome among all the possibilities.

By applying logical reasoning, we accurately set our equation for the number of ways to fail and solve for the number of papers needed in the exam setup.

Problem Solving Steps

Effective problem solving often follows a structured approach to dissecting and addressing the given task. Here’s how we approached the exercise using clear methodological steps:

**1. Understanding the Problem:**
First, establish what you need to find. In this case, it is the number of papers, knowing the fail outcomes total 63.

**2. Logical Analysis:**
Break down the problem logically to figure out how to model it mathematically. We realized that each paper results in two outcomes, leading to \(2^n\) total outcomes for 'n' papers, with only one result guaranteeing passing all papers.

**3. Setting the Equation:**
Translate logical deductions into an equation. Here, \(2^n - 1 = 63\) represents the fail outcomes.

**4. Solving the Equation:**
Solve it to find 'n.' In our example, adjusting to \(2^n = 64\) guided us to \(n = 6\).

**5. Verification:**
Finally, review and verify the obtained result to assure the solution is correct. Verifying that \(2^n - 1 = 63\) holds true confirms the correctness of our previous steps.

By consistently following such steps, problem-solving becomes more systematic and reliable, leading to success even in complex questions.