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### (a) Problem Interpretation:
You are asked to find how many ways 7 scoops of vanilla ice cream can be distributed among Alice, Bob, and Stacey. This is a **"stars and bars"** problem in combinatorics, where each person can receive 0 or more scoops, and the scoops are identical.

To solve it, you need to find how many ways you can distribute 7 scoops into 3 groups. This is equivalent to solving the equation:
\[
x_1 + x_2 + x_3 = 7
\]
where \(x_1\), \(x_2\), and \(x_3\) are non-negative integers representing how many scoops Alice, Bob, and Stacey get.

The formula for the number of non-negative integer solutions to an equation like this is given by the "stars and bars" theorem:
\[
\text{Number of solutions} = \binom{k + n - 1}{n - 1}
\]
where \(k\) is the number of scoops (7) and \(n\) is the number of people (3).

\[
\binom{7 + 3 - 1}{3 - 1} = \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36
\]

Thus, there are **36 ways** to distribute 7 scoops of vanilla ice cream to Alice, Bob, and Stacey.

### (b) General Formula:
For distributing \(k\) scoops to \(n\) people, the general formula, using the same stars and bars approach, is:
\[
\binom{k + n - 1}{n - 1}
\]
This formula calculates the number of ways to distribute \(k\) identical items to \(n\) distinct recipients where each recipient can receive zero or more items.

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Do you want more details on either part? Feel free to ask!

Here are some follow-up questions to deepen understanding:

1. How does the formula change if each person must receive at least one scoop?
2. What is the combinatorial interpretation of the binomial coefficient \(\binom{k + n - 1}{n - 1}\)?
3. Can this formula be applied if the ice cream scoops were distinguishable instead of identical?
4. How many ways are there to distribute 10 scoops among 4 people using the stars and bars method?
5. What if we were distributing 7 scoops but had 4 people instead of 3?

**Tip:** In combinatorics, problems involving distributing identical objects among distinct groups often utilize the stars and bars method, which simplifies counting possibilities!

How many ways can 7 scoops of vanilla ice cream be distributed to Alice, Bob, and Stacey? Write down an explicit general formula for distributing k scoops to n people.

Learn how to calculate the number of ways to distribute 7 scoops of vanilla ice cream to Alice, Bob, and Stacey using combinatorics and the Stars and Bars Theorem. Explore the explicit general formula for distributing k scoops to n people. Suitable for college level or advanced high school students.

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