Problem 114 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of positive integral solutions of the inequality \(3 x+y+z \leq 30\), is (A) 1115 (B) 1215 (C) 1315 (D) None of these

Step-by-Step Solution

Answer

The correct answer is (B) 1215.

Step 1: Understanding the Problem

We are tasked with finding the number of positive integral solutions to the inequality \(3x + y + z \leq 30\). Here, \(x, y,\) and \(z\) must all be positive integers.

Step 2: Transform into Equation

To use the stars and bars method, we will first transform the inequality \(3x + y + z \leq 30\) into an equation. Introduce a new variable \(t\) such that \(3x + y + z + t = 30\) where \(t \geq 0\).

Step 3: Find the Range for Variable x

Since \(x\) is a positive integer, its smallest value is 1. This means \(3x + y + z \leq 30\) is equivalent to \(x \leq 10\), as if \(x\) were greater than 10, then \(3x\) would exceed 30.

Step 4: Iterate for Each x to Count Solutions

We iterate over each possible \(x\) from 1 to 10, and for each case, we solve the transformed equation for \(y, z, t\). Specifically, for each \(x\), solve for \(y + z + t = 30 - 3x\) as non-negative integers.

Step 5: Application of Stars and Bars

For each solved \(y + z + t = k\), use the stars and bars method to find the number of solutions. The formula for counting the number of non-negative integer solutions for an equation like \(y + z + t = k\) is \(\binom{k+2}{2}\).

Step 6: Sum of Solutions Across x

Calculate the number of solutions for each case from \(x=1\) to \(x=10\) and sum them up. For \(x = k\), solve \(y + z + t = 30 - 3k\) yielding \( \binom{30-3k+2}{2} \).

Step 7: Final Calculation

The total number of solutions is the sum: \( \sum_{k=1}^{10} \binom{32-3k}{2}\). Simplify this by computing each term: \(\binom{29}{2} + \binom{26}{2} + \ldots \). Calculate this to arrive at the answer, choosing the one matching the options.

Key Concepts

Positive Integer Solutions

Positive integer solutions refer to solutions where all the variables involved are positive integers. This means that each variable can only be a whole number greater than zero. In the given problem, we seek solutions for the variables \(x, y,\) and \(z\) such that they satisfy the inequality \(3x + y + z \leq 30\).

The restriction that \(x, y,\) and \(z\) must be positive influences how we solve the inequality:

• Since \(x\), \(y\), and \(z\) must each be at least \(1\), their minimum values directly affect the calculations.
• This is handled by first assuming the smallest possible value for \(x\) and then determining the feasible range for \(y\) and \(z\).

Using these constraints in our calculations ensures that all solutions counted are positive integers, aligning with the problem requirements.

Stars and Bars Method

The Stars and Bars method is an elegant combinatorial technique used to find non-negative integer solutions to equations. It's particularly useful in problems where you distribute indistinguishable items into distinguishable bins.

In problems like \(y + z + t = k\), where \(y, z,\) and \(t\) are non-negative integers, this method helps calculate the number of solutions efficiently:

• Visualize the problem as placing 'stars' (representing units) in bins separated by 'bars'.
• For an equation like \(y + z + t = k\), we are essentially arranging \(k\) stars and \(2\) bars (for the two boundaries between the variables) in a line.
• The formula for calculating the number of different ways to arrange \(n\) stars and \(b\) bars is given by the combination \(\binom{n + b}{b}\).

In the problem at hand, this method tells us how many ways we can assign values to \(y, z,\) and \(t\) that fulfill each equation derived for each value of \(x\).

Non-negative Integer Solutions

Non-negative integers include all whole numbers from zero upward. In the context of solving for non-negative integer solutions, these are solutions to an equation where each variable is a non-negative integer.

In the given exercise, once we fix a value for \(x\), we rewrite the inequality as \(y + z + t = 30 - 3x\). The variables \(y\), \(z\), and \(t\) must all take non-negative integer values, which may include zero.

• This ensures that even when some variables become zero, the equation still holds true.
• Using the Stars and Bars method, non-negative values allow us to include all possible arrangements, expanding the potential solutions and ensuring thorough coverage of each case as \(x\) varies.
• This broadens the search space beyond just strictly positive values, capturing all configurations of \(y, z,\) and \(t\) that satisfy the respective equation for each \(x\).

Understanding the concept of non-negative integer solutions is crucial for ensuring that we count every possible way to fulfill the conditions of the problem without accidentally excluding solutions where one or more variables might be zero.