2021 × 2022 × 2023 × [1
4
×
2
5
×
3
6
×
4
7
×
5
8
×
6
9
× … ×
2017
2020
×
2018
2021
×
2019
2022
×
2020
2023 ].

The expression you've provided is a large product involving two parts. Let's break it down step by step:

1. The first part is:

\[
2021 \times 2022 \times 2023
\]

2. The second part is a long product, which appears to be a sequence of fractions. This pattern seems to pair numbers in a ratio, from 1/4 to 2020/2023. The second part can be written as:

\[
\frac{1}{4} \times \frac{2}{5} \times \frac{3}{6} \times \cdots \times \frac{2020}{2023}
\]

Now, the key to simplifying this is to recognize that many terms will cancel out.

### Simplifying the second part:

The sequence \(\frac{1}{4}, \frac{2}{5}, \frac{3}{6}, \dots, \frac{2020}{2023}\) cancels out all the terms in the numerator except the very first and all terms in the denominator except the last few. Let's write this out:

The numerator will be:
\[
1 \times 2 \times 3 \times \cdots \times 2020 = 2020!
\]

The denominator will be:
\[
4 \times 5 \times 6 \times \cdots \times 2023 = 2023!
\]

Therefore, the second part simplifies to:

\[
\frac{2020!}{2023!}
\]

But, by the properties of factorials:
\[
\frac{2020!}{2023!} = \frac{1}{2021 \times 2022 \times 2023}
\]

### Final Simplification:

Multiplying the first and second parts together:

\[
(2021 \times 2022 \times 2023) \times \frac{1}{2021 \times 2022 \times 2023} = 1
\]

Thus, the value of the entire expression is:

\[
\boxed{1}
\]

Would you like more details or have any questions about this solution?

---

Here are 5 related questions:
1. What are factorials and how are they calculated?
2. Can we generalize this type of product for other similar patterns?
3. How does cancellation of terms work in a large product of fractions?
4. What are some common applications of factorials in mathematics?
5. How does the concept of limits apply to factorials?

**Tip:** When dealing with large products or sequences, always check if terms can cancel each other to simplify the calculation.

2021 × 2022 × 2023 × [1/4 × 2/5 × 3/6 × 4/7 × 5/8 × 6/9 × … × 2017/2020 × 2018/2021 × 2019/2022 × 2020/2023].

Discover how to simplify the expression involving 2021 × 2022 × 2023 and a sequence of fractions from 1/4 to 2020/2023. This step-by-step solution explains factorial simplification and cancellation of terms, suitable for Grades 11-12 or undergraduate students.

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