Explaining the Number of Positive Integral Solutions with Examples

In the field of discrete mathematics and combinatorics, determining the number of positive integer solutions to an equation can be approached in several methods. This article will explore one such problem and provide step-by-step explanations with examples to clarify the methodology.

Consider the equation:

Understanding the Problem

The task at hand is to find the number of positive integral solutions to the equation:

Where n, a1, a2, ..., ar1 are positive integers, and r1 is a specified range.

Approach 1: Summation Method

The first method involves breaking down the equation into a sum of individual solutions. This can be expressed as:

tn[1] The number of solutions will be equal to the sum of all individual solutions:

it Where i r1 to n1

Therefore, the total number of solutions is:

rCr r1Cr... nCrn

Here, rCr, r1Cr, ..., nCrn represent combinations, which are the number of ways to select r items from n items, without repetition and without regard to order.

Approach 2: Introducing a New Variable

The second method involves introducing a new positive variable, ar2, which transforms the original equation into:

n2

Here, n2 is a constant value specified for the new equation. This method simplifies the problem to finding the number of positive integral solutions, which can be calculated as:

n1Cr1n

Where n1 is the total number of variables after introducing the new variable, and r1 is the specified range for the variables.

Comparison and Equivalence

Both methods are equivalent and yield the same result. This can be illustrated with an example:

Example: Simplifying the Equation

Consider the equation:

4a1a2a3 12

The goal is to find the number of positive integer solutions.

Approach 1:

We sum up the individual solutions for different combinations of a1, a2, and a3. For instance, if a1 1, then we have:

1 * a2 * a3 12

The solutions are: (1, 1, 12), (1, 2, 6), (1, 3, 4), (1, 4, 3), (1, 6, 2), (1, 12, 1), and so on.

Approach 2:

We introduce a new variable, a4, such that:

4a1a2a3a4 12

The number of positive integer solutions for this equation is:

4Cr1 3! 4C3 4

These solutions are the same as the solutions obtained from the first approach, confirming the equivalence of both methods.

Conclusion

Both methods provide a systematic approach to determining the number of positive integral solutions to an equation. Whether you use the summation method or introduce a new variable, the results will be consistent.

If you have any doubts or require further clarification, feel free to ask in the comments section.

Related Keywords

Positive Integral Solutions: The number of ways positive integers can satisfy an equation.

Combinatorial Methods: Techniques used in combinatorics to solve problems related to counting and arrangement.

Algebraic Equations: Equations involving variables and coefficients that can be solved to find the values of the variables.