Problem 94 - Algebra for JEE ADVANCED

Question

Number of ways in which Rs. 18 can be distributed amongst four persons such that nobody receives less than Rs. 3 is a. \(4^{2}\) b. \(2^{+}\) c. \(4 !\) d. none of these

Step-by-Step Solution

Answer

There are 84 ways, so the answer is (d) none of these.

Step 1: Understand the Problem

We want to distribute Rs. 18 among four people with each receiving at least Rs. 3. Start by calculating how much is left to distribute after ensuring each person gets Rs. 3.

Step 2: Initial Distribution

Give each person Rs. 3. This totals Rs. 3 x 4 = Rs. 12 distributed initially. Therefore, we have Rs. 18 - Rs. 12 = Rs. 6 left to distribute.

Step 3: Use Combinatorial Method

We need to distribute the remaining Rs. 6 among 4 people with each getting 0 or more. This is a classic combinatorics problem of distributing 'n' identical items into 'r' groups.

Step 4: Apply Stars and Bars Theorem

The formula for distributing 'n' identical items into 'r' different groups is given by \( \binom{n + r - 1}{r - 1} \). Substitute \(n = 6\) and \(r = 4\), we have \( \binom{6 + 4 - 1}{4 - 1} = \binom{9}{3} \).

Step 5: Calculate Combinations

Calculate \( \binom{9}{3} \), which is \( \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \). So, there are 84 ways to distribute the remaining Rs. 6.

Step 6: Review Options

The number of ways, 84, does not match any of the provided options a, b, or c. Therefore the correct answer is "none of these".

Key Concepts

Stars and Bars Method

The "stars and bars" method is a clever technique used in combinatorics to solve problems where we need to distribute identical items into different groups. Imagine we have a number of stars (let's call them items or, in this case, remaining rupees) that we want to distribute between several groups or people. To do this efficiently, we use bars as dividers that create partitions among groups.

This method is wonderfully powerful because it transforms the problem into a much simpler one, where the goal becomes counting different ways to place dividers (bars) between items (stars). If we have \( n \) identical items to distribute and \( r \) groups to fill, we are essentially looking for different sequences of stars and bars.

• We need \( r-1 \) bars to separate these groups.
• In total, we arrange \( n + (r-1) \) symbols (stars plus bars).

The formula for counting these arrangements is given by the binomial coefficient, \( \binom{n + r - 1}{r - 1} \). In our problem about distributing \( Rs. 6 \) among 4 people, stars represent the rupees and bars represent the partitions. Using 3 bars, we partition the available rupees among 4 people, which leads to the calculation \( \binom{9}{3} \).

The stars and bars method simplifies the distribution problem by treating it as an arrangements problem, which is easier to handle with combinatorial math.

Binomial Coefficient

The binomial coefficient is a powerful mathematical tool that counts the number of ways to choose a certain number of items from a larger set, without considering the order of selection. It is represented as \( \binom{n}{k} \) and pronounced as "n choose k." In our problem, the binomial coefficient comes into play when we need to count the different ways to arrange our stars and bars.

The formula for calculating a binomial coefficient is:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Where "!" denotes a factorial, which is the product of all positive integers up to that number. For example, \( 3! = 3 \times 2 \times 1 = 6 \).

Applying this to our problem, we know we have 9 positions to fill (6 stars + 3 bars), and we want to choose 3 positions for the bars, hence \( \binom{9}{3} \). Calculating this gives us \( \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \). Therefore, there are 84 possible ways of distributing the remaining rupees among the four people.

Discrete Mathematics

Discrete mathematics is a fascinating area of math dealing with distinct and separated values, often involving counting and arrangement problems. At its core, it deals with objects that can assume separate, distinct values, which is a foundation for subjects like combinatorics and graph theory.

In the context of our rupee distribution problem, discrete mathematics is applied through combinatorial methods, such as the stars and bars method, to determine different ways we can achieve a desired distribution. Because we are working with whole numbers (in this case, rupees and people), the problems and solutions naturally fall under discrete mathematics.

Some key areas within discrete mathematics include:

• Combinatorics: Study of counting, arrangement, and combination of elements.
• Graph Theory: Exploration of vertices and edges, often used for network problems.
• Number Theory: Focus on integers and their properties.

Each of these areas has its own applications and techniques, but they all share the common ground of dealing with finite and countable sets of objects. In our exercise, the understanding of discrete math principles aids in calculating how to distribute rupees efficiently while meeting constraints like minimum amounts for each person.