Find the number of positive integral solutions of the equation `x+y+z=12.`

Explanation

To find the number of positive integral solutions of the equation \( x + y + z = 12 \), we can use the "stars and bars" theorem from combinatorics. Here’s how we can solve it step by step:

### Step 1: Understand the Problem
We need to find the number of ways to distribute 12 identical objects (the total sum of \( x + y + z \)) into 3 distinct groups (the variables \( x, y, z \)), where each group must contain at least one object since \( x, y, z \) are positive integers.

### Step 2: Adjust for Positive Integers
Since \( x, y, z \) must be positive integers, we can redefine the variables:
Let \( x' = x - 1 \), \( y' = y - 1 \), and \( z' = z - 1 \). This transformation ensures that \( x', y', z' \) are non-negative integers (i.e., \( x', y', z' \geq 0 \)).
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