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The problem asks how many ordered quadruplets \((a, b, c, d)\) exist such that the sum of four whole numbers \(a\), \(b\), \(c\), and \(d\) is 15. Each of these numbers is a whole number, meaning \(a, b, c,\) and \(d\) are non-negative integers.

This is a combinatorial problem that can be solved using the "stars and bars" theorem. Here's the solution:

### Step-by-Step Solution

1. **Define the Equation**: We need to find the number of solutions to
   \[
   a + b + c + d = 15
   \]
   where \(a\), \(b\), \(c\), and \(d\) are whole numbers (non-negative integers).

2. **Apply the Stars and Bars Method**: The number of non-negative integer solutions to an equation of the form \(x_1 + x_2 + \cdots + x_k = n\) is given by the formula
   \[
   \binom{n + k - 1}{k - 1}
   \]
   where \(k\) is the number of variables, and \(n\) is the sum.

3. **Substitute Values**:
   - Here, \(k = 4\) (for \(a\), \(b\), \(c\), and \(d\)).
   - \(n = 15\) (the total sum).

   So we need to calculate
   \[
   \binom{15 + 4 - 1}{4 - 1} = \binom{18}{3}
   \]

4. **Calculate \(\binom{18}{3}\)**:
   \[
   \binom{18}{3} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = \frac{4896}{6} = 816
   \]

### Conclusion

There are **816 ordered quadruplets** \((a, b, c, d)\) such that \(a + b + c + d = 15\).

Would you like a deeper explanation of any steps or further assistance?

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Here are five related questions to explore further:

1. How would the solution change if the numbers \(a\), \(b\), \(c\), and \(d\) had to be positive integers?
2. How many ordered triples \((a, b, c)\) satisfy \(a + b + c = 10\)?
3. What is the stars and bars theorem, and how is it applied in combinatorics?
4. How would you solve for ordered quadruplets if the sum were changed to a number other than 15?
5. What would be the solution if \(a\), \(b\), \(c\), and \(d\) had an additional constraint, like \(a \leq 5\)?

**Tip:** For problems involving the number of ways to achieve a sum with non-negative integers, the stars and bars method is often effective for simplifying calculations.

The sum of four whole numbers a, b, c, and d is 15. How many ordered quadruplets (a, b, c, d) exist?

Learn how to count the number of ordered quadruplets (a, b, c, d) that sum to 15. This problem uses combinatorics and the stars and bars theorem, making it suitable for students in Grades 10-12.

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