Problem 133 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of non-negative integral solutions of \(x_{1}+x_{2}+x_{3}+x_{4} \leq n\) (where \(n\) is a positive integer) is (A) \({ }^{n+4} C_{n}\) (B) \({ }^{n+4} C_{4}\) (C) \({ }^{n+3} C_{3}\) (D) \({ }^{n+3} C_{n}\)

Step-by-Step Solution

Answer

The answer is (B) \({}^{n+4}C_{4}\).

Step 1: Understand the problem as an inequality

We need to find the number of non-negative integral solutions for the inequality \(x_1 + x_2 + x_3 + x_4 \leq n\).

Step 2: Convert the inequality to an equation

To transform the inequality \(x_1 + x_2 + x_3 + x_4 \leq n\) into an equation, introduce a new variable \(x_5\) as follows: \(x_1 + x_2 + x_3 + x_4 + x_5 = n\), where \(x_5\) is a non-negative integer representing the 'slack' in the inequality.

Step 3: Solve the equation using the stars and bars method

The equation \(x_1 + x_2 + x_3 + x_4 + x_5 = n\) needs to be solved for non-negative integers. Using the stars and bars method, the number of solutions is given by the formula \(\binom{n+k-1}{k-1}\), where \(k\) is the number of variables. Here \(k=5\), so the formula is \(\binom{n+5-1}{5-1} = \binom{n+4}{4}\).

Key Concepts

Inequality Transformation

In combinatorics, it's common to encounter inequalities where you're looking for certain solutions. Transforming these inequalities into equations can simplify the problem. This transformation is crucial for problems like finding the number of solutions to \( x_1 + x_2 + x_3 + x_4 \leq n \).

Here's how you can approach it:

• Identify the inequality you are working with, like the one above.
• Introduce a new variable to account for the difference between the two sides of the inequality. In this case, we added \( x_5 \) to convert the inequality \( x_1 + x_2 + x_3 + x_4 \leq n \) into an equation \( x_1 + x_2 + x_3 + x_4 + x_5 = n \).

This new variable, \( x_5 \), is sometimes referred to as a 'slack variable', representing the 'gap' or 'slack' that exists in the inequality. By doing this transformation, you now have an equation that can be solved with the stars and bars method, allowing for a straightforward path to finding the solution.

Stars and Bars Method

The stars and bars method is a classic combinatorial technique used to solve problems involving distributing indistinguishable objects (stars) into distinguishable boxes (bars), often used in problems related to partitioning and combinations.

Here's how it works in the context of our transformed equation \( x_1 + x_2 + x_3 + x_4 + x_5 = n \):

• Think of \( n \) as \( n \) stars you want to place into 5 different groups (represented by \( x_1, x_2, x_3, x_4, \) and \( x_5 \)).
• To divide these stars among the variables, introduce 4 dividers or "bars".
• The total number of objects (stars + bars) you arrange is \( n + 4 \) (since there are 4 bars).
• Using combinations, the formula \( \binom{n + 5 - 1}{5 - 1} \) or simply \( \binom{n + 4}{4} \) calculates the number of ways to arrange these stars and bars.

This method enables us to systematically count every possible distribution of stars into the boxes, giving us the number of non-negative integral solutions to the equation.

Non-Negative Integral Solutions

The idea of non-negative integral solutions in combinatorics refers to solutions of equations where each variable is an integer greater than or equal to zero. These are common in problems involving distribution and arrangement.

For the equation \( x_1 + x_2 + x_3 + x_4 + x_5 = n \), we seek the number of ways \( x_1, x_2, x_3, x_4, \) and \( x_5 \) can be non-negative integers. Non-negative means that zero is included in the set of acceptable solutions.

Important aspects to consider include:

• Each variable \( x_i \) (for \( i = 1 \) to 5) represents a group or partition of the total \( n \).
• The method used, such as stars and bars, helps efficiently calculate how these solutions can be partitioned while respecting the non-negativity constraint.
• Such problems often involve calculating combinations, as with our use of \( \binom{n + 4}{4} \), to find the total number of solutions.

Understanding these concepts can significantly aid in solving a wide array of combinatorial problems that require finding non-negative integer solutions.