Question

How many integer solutions are there for the following equations? (a) \(x+y+z=8\) where \(x \geq 0, y \geq 0,\) and \(z \geq 0\) (b) \(x+y+z+t=18\) where \(x, y, z,\) and \(t\) are each greater than zero (c) \(x+y+z+t=12\) where \(x \geq 1, y \geq 2,\) and \(t \geq 1\)

Short answer

(a) 45 solutions, (b) 680 solutions, (c) 165 solutions.

Step-by-step solution

Understanding Part (a)

We are given the equation \(x + y + z = 8\) with the condition that each variable is a non-negative integer. This is a classic example of a 'stars and bars' problem, where we need to find the number of ways to place 'dividers' between 'stars' to separate the sum into parts.

Applying 'Stars and Bars' to (a)

To solve \(x+y+z=8\), we consider 8 stars for the sum and 2 bars to divide the stars between \(x, y,\) and \(z\). The number of ways to arrange the stars and bars is given by the combination formula: \( \binom{8+2}{2} \).

Solving Part (a)

Calculating the combination, we get \( \binom{10}{2} = \frac{10 \times 9}{2} = 45 \). Thus, there are 45 integer solutions where \(x, y, z \geq 0\).

Understanding Part (b)

In this part, \(x + y + z + t = 18\) with each variable having to be greater than zero. We need to ensure that we start from 1 for each variable, effectively transforming the variables to be non-negative.

Transforming Variables in (b)

To ensure each variable is at least 1, substitute \(x = x' + 1\), \(y = y' + 1\), \(z = z' + 1\), \(t = t' + 1\), leading to \(x'+y'+z'+t' = 14\) (since 4 was subtracted from 18).

Applying 'Stars and Bars' to (b)

Now, find the number of solutions to \(x'+y'+z'+t'=14\) in non-negative integers using 'stars and bars' as explained before, \( \binom{14+3}{3} \).

Solving Part (b)

Calculate \( \binom{17}{3} = \frac{17 \times 16 \times 15}{3 \times 2 \times 1} = 680 \). Thus, there are 680 integer solutions where \(x, y, z, t \) are each greater than zero.

Understanding Part (c)

The equation given is \(x+y+z+t=12\) with \(x \geq 1, y \geq 2, t \geq 1\). We need to convert it into non-negative integer form by transforming each variable as required.

Transforming Variables in (c)

Set \(x = x' + 1\), \(y = y' + 2\), \(t = t' + 1\), so the equation becomes \(x'+y'+z+t' = 8\) (since the transformations subtract 4 from 12).

Applying 'Stars and Bars' to (c)

Calculate the number of solutions for \(x'+y'+z+t'=8\) using 'stars and bars': \( \binom{11}{3} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \).

Solving Part (c)

Therefore, there are 165 integer solutions for the equation \(x+y+z+t=12\) with the given conditions.

Key concepts

Stars and Bars Method

The "Stars and Bars" method is a fascinating and powerful technique used in combinatorics to find the number of ways to distribute identical items into distinct groups. Think of it like arranging stars, which represent items, and bars, which signify the dividers between different groups.
For example, if you need to solve an equation like \( x+y+z=8 \), where each variable represents a non-negative integer, you essentially need to figure out how to place two dividers among eight stars. This is because dividing the stars into sections will determine the values of \( x \), \( y \), and \( z \). Each distinct arrangement corresponds to a unique solution.
Using the stars and bars approach, the number of solutions is given by the combination formula:

• \( \binom{8 + 2}{2} \)

This formula means you choose positions for the bars out of the total number of positions available (stars plus bars). In this case, the number gives us a total of 45 ways to arrange them, providing all the possible solutions to the equation.

Non-negative Integer Solutions

When working with equations that involve finding solutions in the form of integers, a common condition is that all variables should be non-negative. A non-negative integer is simply any whole number that is zero or positive. These are useful when dealing with distributions, such as people, objects, or even arbitrary units.
In problems like \( x+y+z=8 \), simply knowing that each variable is non-negative allows us to apply methods like "stars and bars" without additional complex transformations. However, when certain variables must be greater than zero, transformations are necessary.

• For instance, if every variable in an equation must be at least 1, you would subtract 1 from each variable, transforming the problem to one about non-negative equivalents.
• Transforming \( x = x'+1 \), \( y = y'+1 \), etc., simplifies the task into finding solutions with non-negative integers, like \( x'+y'+z'+t'=14 \).

This step illustrates why understanding and employing non-negative integer transformations can simplify finding viable integer solutions.

Combinations in Combinatorics

Combinations play a crucial role in solving problems within combinatorics, especially when you're tasked to find the ways to combine things or arrange them under given constraints, like stars and bars.
The combination formula, notated as \( \binom{n}{r} \), is read as "n choose r" and calculated using

• \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

where \( n! \) (n factorial) is the product of all positive integers up to \( n \).
In the context of distributing items, such as finding all integer solutions to equations like \( x+y+z = 8 \), combinations tell you how many ways you can pick positions for the dividers in the line-up of stars and bars. The formula's elegance lies in its simplicity and power, allowing us to solve intricate distributions by calculating possible arrangements rather than listing them out laboriously.
By mastering combinations, you unlock the ability to tackle numerous types of partitioning problems efficiently. Whether dividing into parts or selecting groups from a set, understanding combinations is key to mastering these problems.