Find the number of non-negative integral solutions of the equation `x+y+z=7`

Explanation

To find the number of non-negative integral solutions of the equation \(x + y + z = 7\), 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 7 identical objects (stars) into 3 distinct groups (represented by \(x\), \(y\), and \(z\)).

### Step 2: Apply the stars and bars theorem
According to the stars and bars theorem, the number of non-negative integral solutions to the equation \(x_1 + x_2 + ... + x_r = n\) is given by the formula:
\[
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