Problem 57 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In a certain test, \(a_{i}\) students gave wrong answers to at least \(i\) questions where \(i=1,2,3, \ldots, k\). No student gave more than \(k\) wrong answers. The total number of wrong answers given is (A) \(a_{1}+a_{2}+\ldots+a_{k}\) (B) \(a_{1}+a_{2}+\ldots+a_{k-1}\) (C) \(a_{1}+a_{2}+\ldots+a_{k+1}\) (D) None of these

Step-by-Step Solution

Answer

(D) None of these.

Step 1: Understanding the Problem

The problem states that there are students who gave wrong answers on a test. The notation \(a_i\) indicates the number of students who got at least \(i\) questions wrong. We need to calculate the total number of wrong answers given by all students combined.

Step 2: Identify Relevant Formulas

Notice that the key to solving the problem is understanding that each group of \(a_i\) students contributes to an aggregate number of wrong answers. Specifically, \(a_i\) students each have wrong answers for at least \(i\) questions, starting from 1 up to \(k\).

Step 3: Calculate Total Wrong Answers

Consider each \(i\) level: \(a_1\) students contribute at least 1 wrong answer each, \(a_2\) contributes at least 1 more wrong answer (in addition to those counted in \(a_1\)), and so on. Therefore, every additional group \(a_i\) contributes more wrong answers to the total.

Step 4: Summation Logic

To find the total number of wrong answers, notice the overlap between the \(a_i\): each \(a_i\) includes students counted in \(a_{i+1}\), forming a cumulative addition by each level. Thus, the total wrong answers is the sum of \(i\) times \(a_i\) for each level from 1 to \(k\).

Step 5: Evaluate Options

Simplify and evaluate the options given: you realize option \(A\) might initially match the large inclusion, but when considering cumulative contributions as \(i \times a_i\), \(A\) is insufficient, so the actual approach doesn’t fully match up as a function-based formula specified in options.

Step 6: Choose the Correct Option

Since the logical addition \(a_1 + a_2 + \ldots + a_k\) doesn't express the compounded contributions accurately, the direct formula given in choices doesn't single out a correct option. Henceforth, acknowledging enumerative discrepancy concludes with choosing "None of these," since the overlapping count isn’t captured directly in expression.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with the counting, arrangement, and combination of elements in sets. In this problem, we are considering how students distribute their wrong answers across multiple questions. Each group of students, denoted by \(a_i\), represents a combination of students who answered at least \(i\) questions incorrectly. This means we need to account for how students might belong to multiple groups, such as those who gave wrong answers to both \(k\) and \(k-1\) questions. To solve problems in combinatorics, understanding how elements overlap and interact is crucial, as it forms the basis for determining the exact count or arrangement in question. In this context, considering the overlap of students across \(k\) categories is key to finding out the total number of wrong answers.

Logical Reasoning

Logical reasoning involves systematically thinking through problems to arrive at sound conclusions. Here, it involves breaking down each contribution of wrong answers by the \(a_i\) groups and understanding their incremental addition to the question total. The logical approach requires recognizing that each \(a_i\) set represents not a static count, but a progressive set of wrong answers. By logical deduction, one notes that since students are grouped based on the number of questions they got wrong, each successive group \(a_{i+1}\) overlaps with \(a_i\). This means that their wrong answers must be counted incrementally, giving rise to summation logic. Being able to apply logic by mapping out these overlaps lets us properly account for the number of errors without double-counting.

Summation Techniques

Summation techniques are strategies used to add up sequences of numbers in a systematic way. In this situation, we can't just add up the \(a_i\) numbers directly. Instead, we must recognize that each \(a_i\) corresponds to multiple wrong answers, forming a series based on their incremental overlap. The challenge is to realize how each student's responses are counted across different \(i\) groups. For example, if one student wrongs both \(i hinspace = 1\) and \(i hinspace = 2\) questions, their wrongs should be included in both categories. This results in an increase observed when moving from one group to the next. Thus, to find the total, it involves adding products of the form \(i imes a_i\), where \(i\) is the threshold number of incorrect answers for a group. Using cumulative summation means understanding the nuances of these overlaps, not just total counts.