The number of non- negative integral solutions of the equation ` x + y + z = 5 ` is :

Options

A. 20
B. 19
C. 21
D. 25

Answer

To find the number of non-negative integral solutions of the equation \( x + y + z = 5 \), we can use the combinatorial method known as "stars and bars".

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the number of ways to distribute 5 identical objects (stars) into 3 distinct groups (variables \( x, y, z \)). Each variable can take a value of 0 or more (non-negative integers).

2. **Applying the Stars and Bars Theorem**: The formula to find the number of non-negative integral solutions for the equation \( x_1 + x_2 + ... + x_n = p \) is given by:
\[ \binom{p + n - 1}{n - 1} \]
where \( p \) is the total number of objects (5 in our case) and \( n \) is the number of variables (3 in our case).

3. **Identifying Variables**: Here, \( p = 5 \) and \( n = 3 \).

4. **Substituting Values into the Formula**:
\[ \binom{5 + 3 - 1}{3 - 1} = \binom{7}{2} \]

5. **Calculating the Binomial Coefficient**:
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2! \cdot 5!} \]
We can simplify this:
\[ = \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

6. **Conclusion**: The number of non-negative integral solutions of the equation \( x + y + z = 5 \) is \( 21 \).