How many non-negative integer solutions are there to the equation rị + *2 + x3 = 21
Quadratic Equation
When it comes to the concept of polynomial equations, quadratic equations can be said to be a special case. What does solving a quadratic equation mean? We will understand the quadratics and their types once we are familiar with the polynomial equations and their types.
Demand and Supply Function
The concept of demand and supply is important for various factors. One of them is studying and evaluating the condition of an economy within a given period of time. The analysis or evaluation of the demand side factors are important for the suppliers to understand the consumer behavior. The evaluation of supply side factors is important for the consumers in order to understand that what kind of combination of goods or what kind of goods and services he or she should consume in order to maximize his utility and minimize the cost. Therefore, in microeconomics both of these concepts are extremely important in order to have an idea that what exactly is going on in the economy.
![**Problem Statement: Finding Non-Negative Integer Solutions**
**Question:**
How many non-negative integer solutions are there to the equation
\[ x_1 + x_2 + x_3 = 21 \]
**Detailed Explanation:**
This problem asks us to find the number of non-negative integer solutions for the given linear Diophantine equation. In other words, we need to determine the number of ways we can assign values to \( x_1 \), \( x_2 \), and \( x_3 \) such that their sum equals 21.
This type of problem is a classic combinatorial problem that can be solved using the "stars and bars" theorem. The stars and bars theorem is used to determine the number of ways to distribute \( n \) identical items into \( k \) distinct groups.
In this context:
- The "stars" represent the total number of items to be divided, which is 21.
- The "bars" will be used to divide these items into 3 distinct groups (representing \( x_1 \), \( x_2 \), and \( x_3 \)).
The formula to determine the number of ways to place \( n \) items into \( k \) groups is given by the combination:
\[ \binom{n + k - 1}{k - 1} \]
In our case:
- \( n = 21 \) (the total sum we want to achieve)
- \( k = 3 \) (the number of variables or groups)
Thus, the number of solutions is:
\[ \binom{21 + 3 - 1}{3 - 1} = \binom{23}{2} \]
Calculating this combination, we get:
\[ \binom{23}{2} = \frac{23!}{2!(23 - 2)!} = \frac{23 \times 22}{2 \times 1} = 253 \]
Therefore, there are 253 non-negative integer solutions to the equation \( x_1 + x_2 + x_3 = 21 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F00a48d79-f805-418c-a679-1c91878d1d75%2F33a027e6-8cbd-4f39-b751-3681f1da1c98%2F2acythn_processed.png&w=3840&q=75)
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