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The number of solutions (x, y, z) to the equation x-y-z = 25, where x, y, z, are positive integers such that x is less than equal to 40, y is less than equal to 12and z less than equal to 12.?
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The number of solutions (x, y, z) to the equation x-y-z = 25, where x,...
Solution:
To solve this question we need to use the concept of generating functions.
Generating Functions:
Generating functions are a way to convert a sequence of numbers into a polynomial. The coefficients of the polynomial represent the sequence of numbers.
Creating the Generating Function:
To create a generating function for this problem, we need to consider all possible values of x, y, and z that satisfy the given conditions. Since x is less than or equal to 40, y is less than or equal to 12, and z is less than or equal to 12, we can create the following generating functions:
x: 1 + x + x^2 + ... + x^40
y: 1 + y + y^2 + ... + y^12
z: 1 + z + z^2 + ... + z^12
The coefficient of x^k in the first generating function represents the number of ways we can choose x such that the sum of x, y, and z is equal to k + 25. Similarly, the coefficient of y^k in the second generating function represents the number of ways we can choose y such that the sum of x, y, and z is equal to k + 25. Finally, the coefficient of z^k in the third generating function represents the number of ways we can choose z such that the sum of x, y, and z is equal to k + 25.
Multiplying the Generating Functions:
To find the total number of solutions, we need to multiply the three generating functions together and find the coefficient of x^k where k ranges from 25 to 77. This is because the sum of x, y, and z must be equal to 25 + k.
After multiplying the generating functions, we get:
1 + 3y + 6y^2 + ... + 3y^24 + 3y^25 + 4y^26 + ... + 4y^51 + 3y^52 + ... + 3y^74 + y^75
The coefficient of y^k in this polynomial represents the number of solutions to the equation x - y - z = 25 where x, y, and z are positive integers such that x is less than or equal to 40, y is less than or equal to 12, and z is less than or equal to 12.
Calculating the Coefficients:
To calculate the coefficients, we can use the following formula:
coefficient of y^k = [x^k] (1 + x + x^2 + ... + x^40) * (1 + y + y^2 + ... + y^12) * (1 + z + z^2 + ... + z^12)
where [x^k] represents the coefficient of x^k.
We can simplify this formula by only considering the terms that contribute to the coefficient of y^k. For example, when k = 25, the only terms that contribute to the coefficient of y^25 are x^40, y^25, and z^1. Therefore, we can calculate the coefficient of y^25 as follows:
coefficient of y^25 = [x^15] (1 + x + x^2 +
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