For the equation `x + y + z +omega= 19,` the number of positive integral solutions is equal to-

Options

A. the number of ways in which 15 identical things can be distributed among 4 persons.
B. the number of ways in which 19 identical things can be distributed among 4 persons.
C. coefficient of `x^(19)` in `(x^(0) + x^(1) + x^(2) + "…….." +x^(19))^(4)`
D. coefficient of `x^(19)` in `(x + x^(1) + x^(2) + "…….." +x^(19))^(4)`

Answer

To find the number of positive integral solutions for the equation \( x + y + z + \omega = 19 \), we can use the "stars and bars" theorem from combinatorics.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the number of positive integral solutions to the equation \( x + y + z + \omega = 19 \). This means \( x, y, z, \) and \( \omega \) must all be positive integers.

2. **Transforming the Variables**:
Since \( x, y, z, \) and \( \omega \) must be positive integers, we can redefine them to simplify our calculations. Let:
\[ x' = x - 1, \quad y' = y - 1, \quad z' = z - 1, \quad \omega' = \omega - 1 \]
where \( x', y', z', \omega' \) are non-negative integers (i.e., they can be zero).

This transformation gives us:
\[ (x' + 1) + (y' + 1) + (z' + 1) + (\omega' + 1) = 19 \]
Simplifying this, we get:
\[ x' + y' + z' + \omega' = 15 \]

3. **Applying Stars and Bars**:
Now, we need to find the number of non-negative integral solutions to the equation \( x' + y' + z' + \omega' = 15 \). According to the stars and bars theorem, the number of ways to distribute \( n \) identical items (stars) into \( k \) distinct groups (bins) is given by:
\[ \binom{n + k - 1}{k - 1} \]
In our case, \( n = 15 \) (the total we want to achieve) and \( k = 4 \) (the number of variables).

4. **Calculating the Combinations**:
Thus, we need to calculate:
\[ \binom{15 + 4 - 1}{4 - 1} = \binom{18}{3} \]

5. **Calculating \( \binom{18}{3} \)**:
Now we compute \( \binom{18}{3} \):
\[ \binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = \frac{4896}{6} = 816 \]

### Final Answer:
The number of positive integral solutions to the equation \( x + y + z + \omega = 19 \) is **816**.