how many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear?

We are asked to find how many ways there are to distribute seven distinct apples and six distinct pears to three distinct people, with the condition that each person gets at least one pear.

### Step 1: Distribute the Pears
Since there are 6 pears and 3 people, and each person must get at least one pear, we need to find a way to distribute these 6 pears such that each person gets at least one. This is a **"stars and bars"** problem with a constraint.

First, since each person must get at least one pear, we can give 1 pear to each of the 3 people, which guarantees that everyone gets at least one pear. This leaves us with \(6 - 3 = 3\) pears to distribute freely among the 3 people. This becomes a standard stars and bars problem where we distribute 3 identical pears to 3 people with no further restrictions.

The number of ways to do this is given by the stars and bars formula:

\[
\text{Number of ways} = \binom{3 + 3 - 1}{3 - 1} = \binom{5}{2} = 10
\]

So, there are 10 ways to distribute the 6 pears such that each person gets at least one.

### Step 2: Distribute the Apples
Now, for the apples: since there are no restrictions on how the apples are distributed, each of the 7 distinct apples can be given to any of the 3 people. The number of ways to distribute 7 distinct apples to 3 distinct people is simply \(3^7\), since each apple has 3 possible recipients.

\[
\text{Number of ways to distribute apples} = 3^7 = 2187
\]

### Step 3: Total Number of Distributions
To get the total number of ways to distribute the apples and pears while satisfying the given condition, we multiply the number of ways to distribute the pears by the number of ways to distribute the apples:

\[
\text{Total number of distributions} = 10 \times 2187 = 21870
\]

Thus, the total number of ways to distribute the apples and pears is **21,870**.

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Would you like more details on any part of this solution? 

Here are 5 questions that expand on this concept:
1. How does the stars and bars formula work in general for unrestricted distributions?
2. What changes if the apples were also subject to some restrictions, such as each person receiving at least one apple?
3. How would the solution differ if the apples and pears were indistinguishable?
4. What happens if we allow some people to receive no fruit at all?
5. Can you explain how combinations are used in the stars and bars formula?

**Tip:** Whenever dealing with "at least one" conditions in combinatorics, distributing a minimum amount first and using stars and bars for the remainder is a helpful strategy!

How many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear?

Explore how to distribute seven distinct apples and six distinct pears among three people, ensuring each person gets at least one pear. This problem highlights combinatorics, including the stars and bars theorem, and involves distributing distinct objects. Suitable for students in Grades 9-12.

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