Problem 64 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The number of ways in which an examiner can assign 30 marks to 8 questions, giving not less than 2 marks to any question, is: \(\quad\) Online April 22, 2013] (a) \({ }^{30} \mathrm{C}_{7}\) (b) \({ }^{21} C_{8}\) (c) \({ }^{21} C_{7}\) (d) \({ }^{30} \mathrm{C}_{8}\)

Step-by-Step Solution

Answer

The correct answer is choice (c): \(\binom{21}{7}\).

Step 1: Understand the problem

We need to distribute 30 marks among 8 questions with each question receiving at least 2 marks. The problem asks us to figure out how many ways we can do this distribution.

Step 2: Allocate minimum marks

Since each of the 8 questions must receive at least 2 marks, we begin by giving 2 marks to each question. Therefore, we allocate \(8 \times 2 = 16\) marks initially.

Step 3: Calculate remaining marks

After allocating the minimum of 2 marks to each question, we have \(30 - 16 = 14\) marks left to distribute without restrictions.

Step 4: Formulate as a stars and bars problem

The task now is to distribute 14 additional marks to the 8 questions in any way. This is a typical 'stars and bars' problem, where we find the number of ways to distribute \(n = 14\) indistinguishable items (marks) to \(k = 8\) distinct groups (questions). The number of ways to do this is given by the formula \(\binom{n+k-1}{k-1}\).

Step 5: Apply the formula

Using the formula \(\binom{n+k-1}{k-1}\), we substitute \(n = 14\) and \(k = 8\): \[ \text{Number of ways} = \binom{14+8-1}{8-1} = \binom{21}{7} \].

Step 6: Find the correct answer choice

The answer from the calculation is \(\binom{21}{7}\). Check the provided answer choices: - Choice (a): \(\binom{30}{7}\) - Choice (b): \(\binom{21}{8}\) - Choice (c): \(\binom{21}{7}\) - Choice (d): \(\binom{30}{8}\) Thus, the correct answer is choice (c): \(\binom{21}{7}\).

Key Concepts

Binomial Coefficient

The concept of a binomial coefficient, often denoted as \( \binom{n}{k} \), is foundational in combinatorics. It represents the number of ways to choose \( k \) items from a total of \( n \) items without regard to the order of selection.
To calculate a binomial coefficient, we use the formula:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \( n! \) denotes the factorial of \( n \), which is the product of all positive integers up to \( n \).

• For example, \( \binom{5}{2} \) equates to finding the number of ways to choose 2 items from 5, which equals 10.


• The binomial coefficient is symmetric, meaning \( \binom{n}{k} = \binom{n}{n-k} \).

Understanding these coefficients is crucial when solving problems involving combinations and distributions, such as determining the ways to divide marks among questions, given specific constraints.

Stars and Bars

The stars and bars method is a powerful technique in combinatorics for distributing indistinguishable items into distinct groups.
In our problem, we distributed 14 remaining marks (stars) among 8 questions (buckets or groups). Here's how it works:

• The stars represent what you are distributing—in our case, marks.
• The bars serve as dividers between different groups, here the questions.

To calculate the number of distributions, we use the formula:\[\binom{n+k-1}{k-1}\]where \( n \) is the number of indistinguishable items and \( k \) is the number of distinct groups.
Applying this to our scenario—distributing 14 marks to 8 questions—the formula turns into \( \binom{14+8-1}{8-1} = \binom{21}{7} \). This approach simplifies complex distribution tasks by converting them into a simple problem of choosing positions for dividers (bars) among the items (stars).

Discrete Mathematics

Discrete mathematics is a field of mathematics that deals with countable, distinct elements. It includes a wide array of topics like combinatorics, graph theory, and logic.
In this exercise, we are primarily concerned with combinatorics, a critical topic within discrete mathematics, which deals with counting, arrangement, and combination of elements. Some essential areas of discrete mathematics include:

• Understanding structures that are fundamentally discrete rather than continuous.


• Utilizing discrete structures for the analysis of problems in computer science, such as algorithm efficiency and data organization.

By using combinatorial methods like binomial coefficients and stars and bars, discrete mathematics provides efficient techniques for solving and understanding problems that need counting and arrangement, particularly evident in our task of distributing marks among questions.