The Appleton Apple Orchard is selling apples to Peppa. There are 4 different varieties available: Gala, Fuji, Granny Smith, and Honeycrisp. They have 500 Gala, 800 Fuji, 300 Granny Smith, and 400 Honeycrisp apples, making a total of 2000 apples available.
Peppa is going to purchase 1800 apples for her and her family. What is the probability that Peppa chooses to buy 450 Gala, 750 Fuji, 250 Granny Smith, and 350 Honeycrisp apples? Hint: Treat each apple variety as a different group. The order she gets the apples in does not matter.
Questions to Answer:
(a) Show the setup / calculations required to find the probability required. No final answer needed.

The problem involves finding the probability of selecting a specific combination of apples from a larger pool, where the order of selection does not matter. This is a classic example of a **multinomial probability** problem, since we are selecting from different groups (varieties) of apples, and we are interested in a specific distribution of those apples.

Here is the setup:

### Total apples available:
- Gala: 500
- Fuji: 800
- Granny Smith: 300
- Honeycrisp: 400
- **Total apples:** 2000

### Peppa's selection:
- Gala: 450
- Fuji: 750
- Granny Smith: 250
- Honeycrisp: 350
- **Total selection:** 1800

### The probability of choosing this specific selection is given by the **multinomial formula**:

\[
P(\text{specific selection}) = \frac{\binom{500}{450} \binom{800}{750} \binom{300}{250} \binom{400}{350}}{\binom{2000}{1800}}
\]

#### Explanation:
1. \(\binom{n}{k}\) is the **binomial coefficient** or "n choose k", which represents the number of ways to choose \(k\) items from \(n\) items. For each variety of apple, we use this formula to determine the number of ways to choose the desired number of apples.
    - \(\binom{500}{450}\): Ways to choose 450 Gala apples from 500 available.
    - \(\binom{800}{750}\): Ways to choose 750 Fuji apples from 800 available.
    - \(\binom{300}{250}\): Ways to choose 250 Granny Smith apples from 300 available.
    - \(\binom{400}{350}\): Ways to choose 350 Honeycrisp apples from 400 available.

2. \(\binom{2000}{1800}\) represents the total number of ways to choose 1800 apples from the entire pool of 2000 apples.

Thus, the probability is the ratio of the number of favorable combinations (choosing the specific 450 Gala, 750 Fuji, 250 Granny Smith, and 350 Honeycrisp) to the total number of ways to choose any 1800 apples from 2000.

### Simplified formula:
\[
P(\text{specific selection}) = \frac{ \binom{500}{450} \binom{800}{750} \binom{300}{250} \binom{400}{350} }{ \binom{2000}{1800} }
\]

This formula represents the setup for calculating the required probability.

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Would you like further clarification on the steps or more details?

Here are 5 related questions to expand on this:
1. What is the general formula for a multinomial distribution?
2. How do we compute binomial coefficients efficiently for large numbers?
3. Can this setup be used for different types of distributions (e.g., Gaussian or Poisson)?
4. How does this problem change if Peppa chooses fewer than 1800 apples?
5. What are the assumptions behind this type of probability calculation?

**Tip:** The binomial coefficient \(\binom{n}{k}\) can be calculated using factorials: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), but for large numbers, it’s more efficient to use computational tools to avoid overflow errors.

The Appleton Apple Orchard is selling apples to Peppa. There are 4 different varieties available: Gala, Fuji, Granny Smith, and Honeycrisp. They have 500 Gala, 800 Fuji, 300 Granny Smith, and 400 Honeycrisp apples, making a total of 2000 apples available. Peppa is going to purchase 1800 apples for her and her family. What is the probability that Peppa chooses to buy 450 Gala, 750 Fuji, 250 Granny Smith, and 350 Honeycrisp apples? Hint: Treat each apple variety as a different group. The order she gets the apples in does not matter.

Explore how to calculate the probability of Peppa selecting 450 Gala, 750 Fuji, 250 Granny Smith, and 350 Honeycrisp apples from the Appleton Apple Orchard's 2000 apples. This detailed solution uses the multinomial probability formula and covers key concepts like binomial coefficients. Suitable for advanced high school students.

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