Introduction

This article explores the problem of finding positive integer solutions for the equation ( x * y * z 12 ) under the constraints ( x leq 3 ) and ( y * z leq 11 ). We will break down the solution into manageable cases and introduce the generating function method to provide a comprehensive understanding of the problem.

Breaking Down the Problem

To solve the equation ( x * y * z 12 ) with the given constraints, we start by considering different possible values for ( x ).

Case 1: ( x 1 )

If ( x 1 ), the equation simplifies to:

1 * y * z 12 implies ( y * z 11 )

The number of positive integer solutions to ( y * z 11 ) can be determined using the formula ( (n - 1) ), where ( n ) is the total sum. Here, ( n 11 ) so:

( text{Solutions} 11 - 1 10 )

Case 2: ( x 2 )

If ( x 2 ), the equation becomes:

2 * y * z 12 implies ( y * z 10 )

Using the same formula, the number of solutions to ( y * z 10 ) is:

( text{Solutions} 10 - 1 9 )

Case 3: ( x 3 )

If ( x 3 ), the equation simplifies to:

3 * y * z 12 implies ( y * z 9 )

Again using the formula, the number of solutions to ( y * z 9 ) is:

( text{Solutions} 9 - 1 8 )

Total Solutions

Adding up the solutions from all three cases:

10 9 8 27

Thus, the total number of positive integer solutions for the equation ( x * y * z 12 ) with the given constraints is 27.

Using Generating Functions

The problem can also be approached using the method of generating functions. The appropriate generating function for this problem, given the constraints, is:

( F(u) (u u^2 u^3)(u u^2 dots u^{11})(u u^2 dots u^{11}) )

We observe that after multiplying out the terms within the brackets, the powers of ( u ) run consecutively from 3 up to 25. The coefficient of ( u^j ) for ( j 3 ) through 25 gives exactly the number of positive integer solutions of the equations ( xyz j ) because the coefficient of each term in the brackets is 1.

We can either use brute force to expand the polynomial of degree 25 or we can sum up the geometric progressions in each bracket and then multiply. The expanded polynomial is given by:

( F(u) frac{(u u^3 - 1)(u u^{11} - 1)^2}{(1 - u)^3} )

This simplifies to the following polynomial, written in ascending order of degrees, where each term ( u^i ) is represented by its coefficient:

( 0, 0, 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 31, 30, 27, 24, 21, 18, 15, 12, 9, 6, 3, 1 )

From the above polynomial, the coefficient of the monomial ( u^{12} ) is 27, indicating that ( xyz 12 ) can arise in 27 ways. This method is superior as it is applicable in all cases involving different numbers of variables, specific sums, and constraints of the type discussed.

Conclusion

The required number of solutions for the problem ( x * y * z 12 ) under the given constraints is 27. The actual solutions can be written out in detail at leisure.