Number of positive unequal integral solutions of the equation `x+y+z=12` is

Options

A. 21
B. 42
C. 63
D. 84

Explanation

To find the number of positive unequal integral solutions of the equation \( x + y + z = 12 \), we can follow these steps:

### Step 1: Understand the problem
We need to find the number of positive integer solutions for the equation \( x + y + z = 12 \) where \( x \), \( y \), and \( z \) are all different (unequal).

### Step 2: Set up the equation
Since \( x \), \( y \), and \( z \) must be positive integers, we can rewrite the equation as:
\[ y + z = 12 - x \]
This means that for each value of \( x \), \( y \) and \( z \) must sum to \( 12 - x \).

### Step 3: Choose values for \( x \)
We will consider values for \( x \) starting from 1 up to 10 (since \( y \) and \( z \) must also be positive integers).

### Step 4: Calculate pairs \( (y, z) \)
For each value of \( x \), we will find pairs \( (y, z) \) such that \( y + z = 12 - x \) and \( y \neq z \).

1. **If \( x = 1 \)**:
\[ y + z = 11 \]
Possible pairs: (2, 9), (3, 8), (4, 7), (5, 6) → 4 pairs

2. **If \( x = 2 \)**:
\[ y + z = 10 \]
Possible pairs: (3, 7), (4, 6) → 3 pairs

3. **If \( x = 3 \)**:
\[ y + z = 9 \]
Possible pair: (4, 5) → 1 pair

4. **If \( x = 4 \)**:
\[ y + z = 8 \]
No valid pairs since \( y \) and \( z \) must be unequal.

5. **If \( x = 5 \)**:
\[ y + z = 7 \]
No valid pairs since \( y \) and \( z \) must be unequal.

6. **If \( x = 6 \)**:
\[ y + z = 6 \]
No valid pairs since \( y \) and \( z \) must be unequal.

7. **If \( x = 7 \)**:
\[ y + z = 5 \]
No valid pairs since \( y \) and \( z \) must be unequal.

8. **If \( x = 8 \)**:
\[ y + z = 4 \]
No valid pairs since \( y \) and \( z \) must be unequal.

9. **If \( x = 9 \)**:
\[ y + z = 3 \]
No valid pairs since \( y \) and \( z \) must be unequal.

10. **If \( x = 10 \)**:
\[ y + z = 2 \]
No valid pairs since \( y \) and \( z \) must be unequal.

### Step 5: Count the valid pairs
From the calculations above, we have:
- For \( x = 1 \): 4 pairs
- For \( x = 2 \): 3 pairs
- For \( x = 3 \): 1 pair

Total pairs = \( 4 + 3 + 1 = 8 \).

### Step 6: Arrange the solutions
Each of these pairs can be arranged in \( 3! \) (6) ways since the order of \( x, y, z \) matters.

Total arrangements = \( 8 \times 6 = 48 \).

### Final Answer
The number of positive unequal integral solutions of the equation \( x + y + z = 12 \) is **48**.

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