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 "stars and bars" theorem from combinatorics.

### Step-by-Step Solution:

1. **Identify the Problem**: We need to find the number of non-negative integral solutions for the equation \( x + y + z = 5 \). Here, \( x \), \( y \), and \( z \) can take any non-negative integer values.

2. **Understanding the Stars and Bars Theorem**: The stars and bars theorem states that if we want to distribute \( n \) identical objects (stars) into \( r \) distinct boxes (variables), the number of ways to do this is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number of identical objects, and \( r \) is the number of boxes.

3. **Apply the Theorem**: In our case, we have:
- \( n = 5 \) (the total number of units we want to distribute, which is the total on the right side of the equation)
- \( r = 3 \) (the number of variables \( x, y, z \))

Plugging these values into the formula gives:
\[ \binom{5 + 3 - 1}{3 - 1} = \binom{7}{2} \]

4. **Calculate the Binomial Coefficient**: Now we need to calculate \( \binom{7}{2} \):
\[ \binom{7}{2} = \frac{7!}{2!(7-2)!} = \frac{7!}{2! \cdot 5!} \]
This simplifies to:
\[ \frac{7 \times 6}{2 \times 1} = \frac{42}{2} = 21 \]

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

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