Answer the following questions independently of each other
Find the total number of non-negative integral solutions of a+b+c=30.

Options

A. 306
B. 406
C. 396
D. 496

Answer

To find the total number of non-negative integral solutions of the equation \( a + b + c = 30 \), we can use the "stars and bars" theorem from combinatorics. This theorem helps us determine the number of ways to distribute \( n \) identical items (in this case, the total of 30) into \( r \) distinct groups (the variables \( a, b, \) and \( c \)).

### Step-by-Step Solution:

1. **Identify the Variables**:
We have the equation \( a + b + c = 30 \). Here, \( a, b, \) and \( c \) are the variables we need to find solutions for.

2. **Determine the Total**:
The total sum we want is \( n = 30 \).

3. **Count the Variables**:
The number of variables \( r = 3 \) (which are \( a, b, \) and \( c \)).

4. **Apply the Stars and Bars Theorem**:
According to the theorem, the number of non-negative integral solutions can be calculated using the formula:
\[ \text{Number of solutions} = \binom{n + r - 1}{r - 1} \]
Substituting the values we have:
\[ \text{Number of solutions} = \binom{30 + 3 - 1}{3 - 1} = \binom{32}{2} \]

5. **Calculate the Binomial Coefficient**:
The binomial coefficient \( \binom{32}{2} \) is calculated as follows:
\[ \binom{32}{2} = \frac{32!}{2!(32 - 2)!} = \frac{32 \times 31}{2 \times 1} \]

6. **Perform the Multiplication**:
Calculate \( 32 \times 31 = 992 \).

7. **Divide by 2**:
Now divide by 2:
\[ \frac{992}{2} = 496 \]

8. **Final Answer**:
Therefore, the total number of non-negative integral solutions of the equation \( a + b + c = 30 \) is \( 496 \).

### Summary:
The total number of non-negative integral solutions of \( a + b + c = 30 \) is \( 496 \).