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

Options

A. (a) The number of ways `15` identical things can be distributed
among `4` persons.
B. (b) The number of ways `19` identical things can be distributed among
`4` persons.
C. (c) Coefficient of `x^(19)` in `(x^(0) + x^(1) + x^(2) + "………" + x^(19))^(4)`
D. (d) Coefficient of `x^(19)` in `(x + x^(2) + x^(3) + "………" + x^(19))^(4)`

Explanation

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

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the number of positive integral solutions to the equation \( x + y + z + \omega = 19 \). Since we are looking for positive integers, each variable must be at least 1.

2. **Transforming the Variables**: To apply the stars and bars theorem, we can transform the variables to account for the positivity constraint. Let:
\[ x' = x - 1, \quad y' = y - 1, \quad z' = z - 1, \quad \omega' = \omega - 1 \]
This transformation implies that \( x', y', z', \omega' \geq 0 \).

3. **Rewriting the Equation**: Substituting the transformed variables into the original equation gives:
\[ (x' + 1) + (y' + 1) + (z' + 1) + (\omega' + 1) = 19 \]
Simplifying this, we have:
\[ x' + y' + z' + \omega' + 4 = 19 \]
Therefore, we can rewrite it as:
\[ x' + y' + z' + \omega' = 15 \]

4. **Applying the Stars and Bars Theorem**: 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 solutions is given by:
\[ \binom{n + r - 1}{r - 1} \]
where \( n \) is the total number (15 in this case) and \( r \) is the number of variables (4 here).

5. **Calculating the Combinations**: Substituting \( n = 15 \) and \( r = 4 \) into the formula gives:
\[ \binom{15 + 4 - 1}{4 - 1} = \binom{18}{3} \]

6. **Final Result**: Therefore, the number of positive integral solutions to the equation \( x + y + z + \omega = 19 \) is:
\[ \binom{18}{3} \]