Problem 77 - Algebra for JEE ADVANCED

Question

The number of integral solutions of \(x+y+z=0\) with \(x \geq-5\), \(y \geq-5, z \geq-5\) is a. 134 b. 136 c. 138 d. 140

Step-by-Step Solution

Answer

The number of solutions is 136.

Step 1: Understanding the Problem

We need to find the total number of integral solutions for the equation \(x + y + z = 0\) where each variable \(x, y, z\) is greater than or equal to -5.

Step 2: Substitute Variables

Introduce new variables where \(x = x' - 5\), \(y = y' - 5\), and \(z = z' - 5\). The condition \(x, y, z \geq -5\) translates to the condition \(x', y', z' \geq 0\).

Step 3: Rewrite the Equation

Substitute the new variables into the equation: \((x' - 5) + (y' - 5) + (z' - 5) = 0\), which simplifies to \(x' + y' + z' = 15\).

Step 4: Find Non-Negative Integer Solutions

The problem now is to find the number of non-negative integer solutions to \(x' + y' + z' = 15\). This can be done using the stars and bars method, which gives us the formula for the number of solutions: \(\binom{n+k-1}{k-1}\), where \(n=15\) and \(k=3\).

Step 5: Calculate the Binomial Coefficient

Applying the stars and bars formula, we calculate: \[\binom{15+3-1}{3-1} = \binom{17}{2} = \frac{17 \times 16}{2} = 136.\]

Step 6: Conclusion

The number of integral solutions for the given constraints is 136.

Key Concepts

Stars and Bars Method

The stars and bars method is a powerful combinatorial technique used to find the number of ways to distribute indistinguishable objects ("stars") into distinguishable bins ("bars"). This method helps in solving problems involving non-negative integer solutions for equations.

The general idea is simple: if we want to distribute \( n \) identical items into \( k \) distinct groups, we place \( n \) stars in a row and then insert \( k-1 \) bars to divide the stars into groups. This creates sections, each representing a group with a number of stars.

For instance, if we have the equation \( x' + y' + z' = 15 \), and we want non-negative solutions (where each variable represents a group), we can represent this scenario with stars and bars. Here, \( n = 15 \) stars, and we need \( k-1 = 2 \) bars, as we are dividing the stars among three variables.

By using this approach, we can efficiently count the number of valid distributions, which leads us to the use of binomial coefficients.

Binomial Coefficient

Binomial coefficients are crucial in combinatorics for counting combinations. In general, they describe the number of ways to choose \( k \) items from a set of \( n \) items, regardless of the order. The binomial coefficient is denoted by \( \binom{n}{k} \). They appear in the expansion of binomials and in combinatorial methods such as the stars and bars.

In the problem of finding non-negative integer solutions to \( x' + y' + z' = 15 \), the formula to calculate the number of solutions is given by the binomial coefficient \( \binom{n+k-1}{k-1} \). This formula arises because we need to choose where to place the bars among the stars. Here, \( n \) represents the total stars, and \( k \) is the number of variables.

Plugging in our values, we get \(\binom{15+3-1}{3-1} = \binom{17}{2} \). This coefficient tells us there are 136 ways to solve the equation with non-negative integers, making it the answer for this particular exercise.

Non-Negative Integer Solutions

Non-negative integer solutions for equations like \( x' + y' + z' = 15 \) involve finding combinations of integers that satisfy the equation, where each integer is zero or positive. These types of solutions are often found in problems involving allocations, distributions, and partitioning of resources.

In this exercise, using the method of non-negative solutions transforms the original equation \( x + y + z = 0 \) under the condition \( x, y, z \geq -5 \) to another equation \( x' + y' + z' = 15 \) where \( x', y', z' \) are each greater than or equal to zero. This transformation is achieved by substituting \( x = x' - 5 \), \( y = y' - 5 \), and \( z = z' - 5 \).

The significance of this transformation is that it uses known methods (like stars and bars and binomial coefficients) to efficiently calculate the number of solutions without manually listing them. By utilizing the combination formula, we harness mathematical tools to solve what appears at first to be a complex problem in a straightforward manner.