Problem 22
Question
Find the number of solutions to each equation, where the variables are nonnegative integers. $$x_{1}+x_{2}+x_{3}=3$$
Step-by-Step Solution
Verified Answer
There are 10 distinct solutions to the given equation \(x_1 + x_2 + x_3 = 3\) where the variables are non-negative integers.
1Step 1: Analyze the problem
Let the three variables be \(x_1\), \(x_2\), and \(x_3\). The equation given is:
\(x_1 + x_2 + x_3 = 3\)
We are asked to find the total number of ways to distribute the 3 units among these three variables so that the equation holds true while keeping all the variables as non-negative integers.
2Step 2: Use the formula for combinations with repetitions
The formula for combinations with repetitions is given by:
\(\textrm{Number of combinations} = \dbinom{n+r-1}{r-1}\)
Where \(n\) is the number of types of objects (in this case, the 3 units to distribute) and \(r\) is the number of objects in each selection (in this case, the 3 variables to distribute the units among).
3Step 3: Calculate the number of combinations
In this case, we have \(n = 3\) (the 3 units to distribute) and \(r = 3\) (the 3 variables to distribute the units among). Now, plug these values into the combination formula:
\(\dbinom{n+r-1}{r-1} = \dbinom{3+3-1}{3-1} = \dbinom{5}{2}\)
Now, calculate the binomial coefficient:
\(\dbinom{5}{2} = \dfrac{5!}{2!(5-2)!} = \dfrac{5!}{2!3!} = \dfrac{5 \times 4}{2} = 10\)
So, there are 10 distinct solutions to the given equation where the variables are non-negative integers.
Key Concepts
Non-negative Integer SolutionsBinomial CoefficientDiscrete Mathematics
Non-negative Integer Solutions
When faced with problems in combinatorics, especially those involving equations and variables, finding non-negative integer solutions is a common task. These are solutions where the variables can take any whole number value starting from zero up to infinity. In our example, the equation is \(x_1 + x_2 + x_3 = 3\).
The problem asks us to distribute three units (which could be any item or abstract quantity, such as apples or points) across three different variables. The key thing to remember is that each variable can hold a value from zero to three, as long as they all add up to three. Imagine you have three compartments to fill with three identical balls. A compartment can hold all three balls or none at all, or any combination in between. The exercise of finding solutions to such problems is not only fascinating but also a stepping stone to understanding more complex concepts in discrete mathematics.
The problem asks us to distribute three units (which could be any item or abstract quantity, such as apples or points) across three different variables. The key thing to remember is that each variable can hold a value from zero to three, as long as they all add up to three. Imagine you have three compartments to fill with three identical balls. A compartment can hold all three balls or none at all, or any combination in between. The exercise of finding solutions to such problems is not only fascinating but also a stepping stone to understanding more complex concepts in discrete mathematics.
Binomial Coefficient
In combinatorics, the binomial coefficient is a fundamental mathematical concept expressed as \(\dbinom{n}{k}\), which represents the number of ways to choose k elements out of a set of n distinct elements without regard to the order. This concept is essential when we deal with combinations or subsets of a set. In the problem of distributing the three units among our variables, we're essentially looking for the binomial coefficient \(\dbinom{5}{2}\).
The calculation of this coefficient involves factorials, where \(n!\) denotes the product of all positive integers up to n. Specifically, \(\dbinom{5}{2}\) can be computed as follows:
\[\dbinom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2} = 10\]
This tells us there are 10 unique combinations to place our units across the variables. It's like figuring out how many different pairs you can form from a group of five people.
The calculation of this coefficient involves factorials, where \(n!\) denotes the product of all positive integers up to n. Specifically, \(\dbinom{5}{2}\) can be computed as follows:
\[\dbinom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2} = 10\]
This tells us there are 10 unique combinations to place our units across the variables. It's like figuring out how many different pairs you can form from a group of five people.
Discrete Mathematics
The field of discrete mathematics is of particular interest to computer science and information theory. Discrete mathematics deals with distinct or separated values, rather than continuous values. Think of it as the mathematics of things that can be listed or counted, like integers, graphs, or logical statements.
Problems involving non-negative integer solutions or the calculation of binomial coefficients fall under the vast umbrella of discrete mathematics. The applications are far-reaching, from algorithms, to network design, to cryptography, and even to the organization of data in databases.
In our problem, we're using principles from discrete mathematics to explore possible allocations or selections, which can also be described as a form of enumerative combinatorics – counting the number of ways certain patterns can be formed. Understanding such concepts is not just a theoretical exercise; it provides a foundation for solving real-world issues in various fields of computer science and engineering.
Problems involving non-negative integer solutions or the calculation of binomial coefficients fall under the vast umbrella of discrete mathematics. The applications are far-reaching, from algorithms, to network design, to cryptography, and even to the organization of data in databases.
In our problem, we're using principles from discrete mathematics to explore possible allocations or selections, which can also be described as a form of enumerative combinatorics – counting the number of ways certain patterns can be formed. Understanding such concepts is not just a theoretical exercise; it provides a foundation for solving real-world issues in various fields of computer science and engineering.
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