Problem 152 - A Complete Resource Book in Mathematics for JEE Main

Question

Instructions: In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: In an examination consisting of 9 papers, a candidate has to pass in more papers than the numbers of papers in which he fails in order to be successful. The number of ways in which he can be unsuccessful 256 . Reason: \({ }^{n} C_{0}+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots+{ }^{n} C_{n}=2^{n}\)

Step-by-Step Solution

Answer

(A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A).

Step 1: Understanding the Assertion

The assertion states that to be successful, a candidate must pass in more than half of the 9 papers, meaning he needs to pass in at least 5 papers. The candidate is unsuccessful if he passes in 0 to 4 papers.

Step 2: Calculate the Ways to be Unsuccessful

To find the number of ways the candidate can be unsuccessful, calculate the combinations of passing 0 to 4 papers using the formula for combination: \[{\sum_{k=0}^{4}}^{{9} C_{k}} = {9} C_{0} + {9} C_{1} + {9} C_{2} + {9} C_{3} + {9} C_{4}.\]

Step 3: Validation of Reason with Combination Formula

The reason provides the formula \(^{n} C_{0}+^{n} C_{1}+^{n} C_{2}+dots+^{n} C_{n}=2^{n}\). Using this, for \(n=9\), the total number of ways to pass any number of papers (including 5 to 9) is \(2^9\).

Step 4: Calculate Unsuccessful Scenarios Using Reason

Using the relation from the reason, if total scenarios to pass are \(2^9=512\), then unsuccessful scenarios are given by subtracting the count of passing 5 to 9 papers. Thus, unsuccessful scenarios are: \[^{9} C_{0}+^{9} C_{1}+^{9} C_{2}+^{9} C_{3}+^{9} C_{4} = 256.\]

Step 5: Conclusion

Since the number of unsuccessful scenarios calculated aligns with the assertion and uses the given reason, the assertion is true, the reason is true, and the reason correctly explains the assertion.

Key Concepts

Binomial Theorem

The binomial theorem is a fundamental principle in combinatorics. It helps us understand how to expand expressions raised to a power. Let's consider an expression \((a + b)^n\). To expand this, we use the binomial theorem, which is represented as: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). This tells us that each term in the expansion consists of products of the two terms \(a\) and \(b\), raised to powers that add up to \(n\).

The coefficients \(\binom{n}{k}\) are "binomial coefficients". These coefficients tell us how many ways the terms can be selected or rearranged.

• \(\binom{n}{k}\) represents the number of combinations where \(k\) elements are chosen from \(n\) elements without considering the order.
• The sum of the coefficients is always equal to \(2^n\) when \(a\) and \(b\) both equal to 1. This is because there are \(2^n\) possible combinations of selecting terms from the expression.
• Understanding this helps solve problems like the one in the exercise, where the sum of combinations up to a certain point reveals useful information.

Problem Solving

Problem solving in mathematics involves breaking down complex problems into simpler, understandable parts. Here’s how you can approach a problem like the one given in the exercise:

• **Understanding the Problem**: Start by clearly understanding the statement. What is needed, and what is given? Here, the task is to find out in how many ways a candidate can fail by passing fewer than half the exams.


• **Using Known Formulas**: We use binomial coefficients to find out the combinations for passing a certain number of exams. These help in calculating possible outcomes, whether concerning success or failure.


• **Applying Logical Steps**: Follow step-by-step methods to logically arrive at the desired conclusion, ensuring each step is consistent with the previous one.
• **Validation**: Verify that your solution makes sense in the context of the problem. Check if the numbers add up and make logical sense.

In these exercises, it's crucial to not only look for the correct numbers but also to understand the process of arriving at those numbers.

Logical Reasoning

Logical reasoning is an essential part of solving any problem in mathematics or elsewhere. It involves using known principles to arrive at a conclusion. In this exercise, logical reasoning plays a key role:

• **Analyzing Conditions**: Understand the conditions under which the candidate is successful or unsuccessful. Here, logical reasoning helps determine that failing more than half of the exams means the candidate is unsuccessful.


• **Using Mathematical Arguments**: We deploy mathematical theorems and principles, like the binomial theorem, to deduce the number of ways a candidate can fail by analyzing all possible combinations.


• **Cross Fact-Checking**: Logical reasoning helps in validating the solutions. You need to constantly check if your conclusions align with known facts or formulas, such as ensuring that cumulative scenarios add up to the expected total number of possibilities.

Logical reasoning aids in simplifying your thought process, ensuring clarity, and helping you see through the mathematical fog to reach the accurate answer.