The total number of non-negative integral solutions of `x_(1)+x_(2)+x_(3)+x_(4)=100`, is

Options

A. `""^(103)C_(3)`
B. `""^(103)C_(4)`
C. `""^(104)C_(3)`
D. `""^(104)C_(4)`

Answer

To find the total number of non-negative integral solutions of the equation \(x_1 + x_2 + x_3 + x_4 = 100\), we can use the combinatorial method known as "stars and bars." Here’s a step-by-step solution:

### Step 1: Understand the Problem
We need to find the number of ways to distribute 100 identical objects (stars) into 4 distinct boxes (variables \(x_1, x_2, x_3, x_4\)), where each box can hold zero or more objects.

### Step 2: Identify the Variables
Let:
- \(n = 100\) (the total number of objects)
- \(r = 4\) (the number of boxes)

### Step 3: Apply the Stars and Bars Theorem
According to the stars and bars theorem, the number of non-negative integral solutions to the equation \(x_1 + x_2 + x_3 + x_4 = n\) is given by the formula:
\[ \binom{n + r - 1}{r - 1} \]
In our case, we substitute \(n = 100\) and \(r = 4\).

### Step 4: Substitute Values into the Formula
Now we calculate:
\[ \binom{100 + 4 - 1}{4 - 1} = \binom{103}{3} \]

### Step 5: Calculate \(\binom{103}{3}\)
To compute \(\binom{103}{3}\):
\[ \binom{103}{3} = \frac{103 \times 102 \times 101}{3 \times 2 \times 1} \]

### Step 6: Perform the Calculation
Calculating the numerator:
\[ 103 \times 102 = 10506 \]
\[ 10506 \times 101 = 1061106 \]
Now, divide by the denominator:
\[ \frac{1061106}{6} = 176851 \]

### Final Answer
Thus, the total number of non-negative integral solutions of the equation \(x_1 + x_2 + x_3 + x_4 = 100\) is:
\[ \boxed{176851} \]