Problem 153 - A Complete Resource Book in Mathematics for JEE Main

Question

A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is (A) 140 (B) 196 (C) 280 (D) 346

Step-by-Step Solution

Answer

There are 196 different ways to choose the questions.

Step 1: Understand the Problem

The student needs to choose 10 questions out of 13, and he must choose at least 4 questions from the first 5.

Step 2: Determine Possible Scenarios

Identify possible selections by choosing 4, 5, or more from the first five and filling in the rest from the remaining questions. The options are to choose 4 or 5 from the first five questions.

Step 3: Calculate Scenario 1 (Choose 4 from First 5)

If the student chooses 4 from the first 5 questions, he needs 6 more questions from the remaining 8. Calculate the combinations as follows: \( \binom{5}{4} \times \binom{8}{6} \).

Step 4: Calculate Scenario 1 Using Combinatorics

Calculate the number of ways to choose 4 out of 5: \( \binom{5}{4} = 5 \). Calculate the number of ways to choose 6 out of 8: \( \binom{8}{6} = 28 \). Thus the total ways for this scenario is \( 5 \times 28 = 140 \).

Step 5: Calculate Scenario 2 (Choose 5 from First 5)

If the student chooses all 5 from the first group, he needs 5 more from the remaining 8 questions. Calculate \( \binom{5}{5} \times \binom{8}{5} \).

Step 6: Calculate Scenario 2 Using Combinatorics

Calculate the number of ways to choose all 5 from 5: \( \binom{5}{5} = 1 \). Calculate the number of ways to choose 5 out of 8: \( \binom{8}{5} = 56 \). Thus the total for this scenario is \( 1 \times 56 = 56 \).

Step 7: Add Results from Both Scenarios

Add the possibilities from both scenarios: 140 (from choosing 4) + 56 (from choosing 5) = 196.

Key Concepts

Binomial Coefficient

A binomial coefficient is a way to describe the number of ways to choose a subset of items from a larger set, without considering the order. It is widely used in combinatorics to calculate combinations. The notation for a binomial coefficient is \( \binom{n}{k} \), which reads "n choose k".

• Here, \( n \) represents the total number of items.
• \( k \) denotes the number of items to choose.

To compute \( \binom{n}{k} \), the formula \( \frac{n!}{k! \, (n-k)!} \) is used, where \(!\) symbolizes factorial, a product of all positive integers up to that number.
For instance, in our problem, calculating \( \binom{5}{4} = 5 \) and \( \binom{8}{6} = 28 \) helps determine the possible combinations of questions the student can choose. Mastering binomial coefficients is fundamental in solving problems involving combinations and permutations.

Problem Solving in Mathematics

Problem-solving in mathematics is a logical process, employing strategies and methods to find a solution. In the context of our exercise, the problem was dividing the task into smaller scenarios. Here, it was broken into two possible scenarios based on the fixed criteria:

• Choosing at least 4 questions from the first 5.
• Filling the rest from the remaining questions.

This highlights a classic strategy—breaking problems into manageable parts or cases. By identifying different possible cases and computing each separately, you develop a clearer path to the solution. Once this breakdown is performed, it is much easier to apply the correct combinatorial strategies for each scenario. Problem-solving effectiveness in mathematics is often improved through such structured approaches.

Mathematical Reasoning

Mathematical reasoning involves logical thinking to analyze problems and select appropriate methods and formulas. In our scenario, it is employed to discern how to allocate the choice among different groups of questions logically.
The student utilized reasoning to understand and apply the constraints of the problem. The must-choose condition from the first five questions enforced a thoughtful approach:

• Calculate the combinations if choosing 4 questions out of the first 5.
• Calculate the combinations if choosing all 5 questions from these five.

Effective mathematical reasoning relies on understanding both the problem requirements and the applicable mathematical tools, like binomial coefficients in this case. The reasoning not only guides calculations but also validates them, ensuring a coherent resolution is achieved when the results from each scenario are consolidated for the final answer.