This problem can be solved using the stars and bars method, which is a common technique for distributing identical objects (stars) among distinct groups (bars).

Problem Breakdown:

We need to distribute 10 identical apples among 4 people, and each person must receive at least one apple. This means we must first give one apple to each person to satisfy the condition that everyone gets at least one apple. After distributing 1 apple to each of the 4 people, we are left with:

$10 - 4 = 6 \text{ apples}$

Now, we need to distribute these remaining 6 apples among the 4 people with no restrictions (they can receive 0 or more apples). This is now a classic stars and bars problem.

Stars and Bars Formula:

The general formula for distributing $n$ identical objects into $k$ distinct groups is given by:

$\binom{n + k - 1}{k - 1}$

Where $n$ is the number of objects (apples in this case) and $k$ is the number of groups (people).

Here, we need to distribute 6 apples among 4 people, so we plug into the formula:

$\binom{6 + 4 - 1}{4 - 1} = \binom{9}{3}$

Now, calculate $\binom{9}{3}$:

$\binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = \frac{504}{6} = 84$

Final Answer:

There are 84 ways to distribute 10 identical apples among 4 people such that each person gets at least one apple.

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Would you like any further details or clarification? Here are some related questions:

1. How would the problem change if the number of apples or people increased?
2. What if there were no restriction on how many apples each person gets?
3. How does the stars and bars method apply to other types of combinatorics problems?
4. What if some people could receive zero apples?
5. How can generating functions be used to solve similar distribution problems?

Tip: When using the stars and bars method, always remember to adjust for initial conditions (like giving everyone at least one apple) before applying the formula!