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^(2)+x^(3)+ . .. +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 in combinatorics. Here’s a step-by-step solution:

### Step 1: Understand 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.

### Step 2: Transform the Variables
To convert the problem into one involving non-negative integers, we can substitute each variable:
- Let \( x' = x - 1 \)
- Let \( y' = y - 1 \)
- Let \( z' = z - 1 \)
- Let \( \omega' = \omega - 1 \)

This means \( x', y', z', \omega' \) are non-negative integers (i.e., \( x', y', z', \omega' \geq 0 \)). The original equation transforms to:
\[ (x' + 1) + (y' + 1) + (z' + 1) + (\omega' + 1) = 19 \]
which simplifies to:
\[ x' + y' + z' + \omega' = 15 \]

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

### Step 4: Calculate the Number of Solutions
Substituting into the formula:
\[ \binom{15 + 4 - 1}{4 - 1} = \binom{18}{3} \]

### Step 5: Compute \( \binom{18}{3} \)
Now we calculate \( \binom{18}{3} \):
\[ \binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = \frac{4896}{6} = 816 \]

### Conclusion
Thus, the number of positive integral solutions to the equation \( x + y + z + \omega = 19 \) is \( 816 \).

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