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The image contains a combinatorics problem asking how many solutions exist for the equation \( x + y + z + w = 15 \), where \( x, y, z, w \in \mathbb{N}_0 \), meaning that they are non-negative integers.

This type of problem is solved using the *stars and bars* theorem, which gives the number of ways to distribute \( n \) identical items into \( k \) distinct groups, where each group can contain zero or more items.

### Problem setup:
We need to solve the equation \( x + y + z + w = 15 \) in non-negative integers, which is equivalent to distributing 15 identical "stars" into 4 groups (corresponding to \( x \), \( y \), \( z \), and \( w \)).

### Formula:
The number of solutions to the equation \( x_1 + x_2 + \cdots + x_k = n \), where \( x_1, x_2, \ldots, x_k \in \mathbb{N}_0 \), is given by:
\[
\text{Number of solutions} = \binom{n+k-1}{k-1}
\]
In this case, \( n = 15 \) and \( k = 4 \), so the number of solutions is:
\[
\binom{15+4-1}{4-1} = \binom{18}{3}
\]

Now, I'll compute \( \binom{18}{3} \).The number of solutions to the equation \( x + y + z + w = 15 \) in non-negative integers is \( \binom{18}{3} = 816 \).

Thus, the correct answer is \( \boxed{816} \).

Would you like further details or have any additional questions?

Here are five related questions you might find interesting:
1. How would the solution change if some variables were restricted to positive integers?
2. What is the stars and bars theorem, and why does it apply here?
3. How would the solution differ if there were only 3 variables?
4. Can this method be used for equations with upper bounds on the variables?
5. How does this approach relate to partitions of integers?

**Tip:** In combinatorics, problems involving distributing objects into bins often use the "stars and bars" theorem, which is key when dealing with non-negative integer solutions.

How many solutions (x, y, z, w) ∈ N^0 of non-negative integers are there for the equation x + y + z + w = 15?

Explore how to find the number of non-negative integer solutions to the equation x + y + z + w = 15 using combinatorics. Understand the stars and bars theorem, the key mathematical concept applied here. Suitable for Grades 10-12.

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