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

Options

A. 405
B. 406
C. 303
D. 496

Answer

To find the total number of positive integral solutions of the equation \( a + b + c = 30 \), we can use the combinatorial method known as "stars and bars". Here are the steps to solve the problem:

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the number of positive integer solutions to the equation \( a + b + c = 30 \). Since \( a, b, c \) must be positive integers, we can rewrite each variable as \( a' = a - 1 \), \( b' = b - 1 \), and \( c' = c - 1 \). This transformation ensures that \( a', b', c' \) are non-negative integers.

2. **Transforming the Equation**: By substituting \( a, b, c \) with \( a', b', c' \), the equation becomes:
\[ (a' + 1) + (b' + 1) + (c' + 1) = 30 \]
Simplifying this gives:
\[ a' + b' + c' + 3 = 30 \]
Therefore:
\[ a' + b' + c' = 27 \]

3. **Applying the Stars and Bars Theorem**: We now need to find the number of non-negative integer solutions to the equation \( a' + b' + c' = 27 \). According to the stars and bars theorem, the number of solutions is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number we want to achieve (27) and \( r \) is the number of variables (3 in this case).

4. **Substituting Values into the Formula**: Here, \( n = 27 \) and \( r = 3 \). Thus, we need to calculate:
\[ \binom{27 + 3 - 1}{3 - 1} = \binom{29}{2} \]

5. **Calculating the Binomial Coefficient**: The binomial coefficient \( \binom{29}{2} \) is calculated as follows:
\[ \binom{29}{2} = \frac{29!}{2!(29 - 2)!} = \frac{29 \times 28}{2 \times 1} = \frac{812}{2} = 406 \]

6. **Conclusion**: Therefore, the total number of positive integral solutions to the equation \( a + b + c = 30 \) is \( 406 \).