The number of ordered triplets, positive integers which are solutions of the equation x+y+z=100 is:

Options

A. 5081
B. 6005
C. 4851
D. 498

Answer

To find the number of ordered triplets of positive integers \( (x, y, z) \) that satisfy the equation \( x + y + z = 100 \), we can follow these steps:

### Step 1: Understand the Problem
We need to find the number of ordered triplets of positive integers \( x, y, z \) such that their sum equals 100. Since \( x, y, z \) are positive integers, the minimum value for each of them is 1.

### Step 2: Transform the Variables
To simplify the problem, we can redefine the variables:
- Let \( x' = x - 1 \)
- Let \( y' = y - 1 \)
- Let \( z' = z - 1 \)

This transformation ensures that \( x', y', z' \) are non-negative integers (i.e., \( x', y', z' \geq 0 \)).

### Step 3: Rewrite the Equation
Substituting the new variables into the original equation gives:
\[ (x' + 1) + (y' + 1) + (z' + 1) = 100 \]
This simplifies to:
\[ x' + y' + z' + 3 = 100 \]
Thus, we have:
\[ x' + y' + z' = 97 \]

### Step 4: Apply the Stars and Bars Theorem
Now, we need to find the number of non-negative integer solutions to the equation \( x' + y' + z' = 97 \). This is a classic problem that can be solved using the "stars and bars" theorem.

According to the stars and bars theorem, the number of ways to distribute \( n \) indistinguishable objects (stars) into \( r \) distinguishable boxes (variables) is given by:
\[ \binom{n + r - 1}{r - 1} \]
In our case, \( n = 97 \) (the total we want to achieve) and \( r = 3 \) (the three variables \( x', y', z' \)).

### Step 5: Calculate the Number of Solutions
Using the formula, we have:
\[ \text{Number of solutions} = \binom{97 + 3 - 1}{3 - 1} = \binom{99}{2} \]

### Step 6: Compute \( \binom{99}{2} \)
Now, we calculate \( \binom{99}{2} \):
\[ \binom{99}{2} = \frac{99 \times 98}{2 \times 1} = \frac{9702}{2} = 4851 \]

### Conclusion
Thus, the number of ordered triplets \( (x, y, z) \) that satisfy the equation \( x + y + z = 100 \) is \( 4851 \).