https://file.aiquickdraw.com/mathbot/file/17279888435279wso4ppp.jpg

The equation in the image is:

\[
\frac{n!}{(n-2)!} = 20
\]

This simplifies as follows:

1. We know that \( n! \) (n factorial) can be expanded as \( n \times (n-1) \times (n-2)! \). Thus, the factorial expression becomes:

\[
\frac{n!}{(n-2)!} = \frac{n \times (n-1) \times (n-2)!}{(n-2)!}
\]

2. The \( (n-2)! \) cancels out from both the numerator and denominator, leaving:

\[
n \times (n-1) = 20
\]

3. Expanding:

\[
n(n-1) = 20
\]

4. This is a quadratic equation:

\[
n^2 - n - 20 = 0
\]

5. We can solve this using the quadratic formula:

\[
n = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-20)}}{2(1)}
\]
\[
n = \frac{1 \pm \sqrt{1 + 80}}{2}
\]
\[
n = \frac{1 \pm \sqrt{81}}{2}
\]
\[
n = \frac{1 \pm 9}{2}
\]

This gives two solutions:

\[
n = \frac{1 + 9}{2} = 5 \quad \text{or} \quad n = \frac{1 - 9}{2} = -4
\]

Since \( n \) must be positive, the solution is:

\[
n = 5
\]

Thus, \( n = 5 \) satisfies the equation.

Would you like more details or have any questions? Here are some related questions for further exploration:

1. How do you derive the quadratic formula?
2. What are factorials and how do they work in combinations?
3. What happens if \( n! \) appears in more complex expressions?
4. How do you solve more general forms of quadratic equations?
5. What is the importance of canceling terms when simplifying factorial expressions?

**Tip**: When solving factorial-related problems, always try to simplify by canceling common terms before expanding the expressions.

Learn how to solve the factorial equation n! / (n-2)! = 20 with a detailed step-by-step solution. This problem involves simplifying factorial expressions and solving a quadratic equation. Suitable for students in Grades 9-11.

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