The image shows the following expression to compute:

$\frac{n!}{(n - 3)!}$

Solution:

The factorial function $n!$ is defined as the product of all positive integers from 1 to $n$. When we divide two factorials, such as $\frac{n!}{(n - 3)!}$, we can cancel terms to simplify.

1. Start by expanding $n!$: $n! = n \cdot (n-1) \cdot (n-2) \cdot (n-3)!$

2. The $(n - 3)!$ in the denominator will cancel out the $(n - 3)!$ in the numerator, leaving: $\frac{n!}{(n - 3)!} = n \cdot (n-1) \cdot (n-2)$

Thus, the simplified result is:

$n \cdot (n-1) \cdot (n-2)$

Would you like a more detailed explanation, or have any questions about this?

Related Questions:

1. How is the factorial function defined for non-integer values?
2. How would you compute $\frac{n!}{(n - k)!}$ for any $k$?
3. What is the binomial coefficient and how does it relate to factorials?
4. Can you express $n!$ in terms of the Gamma function?
5. How would you apply this formula if $n = 6$?

Tip:

When simplifying factorials, look for common terms in the numerator and denominator to make cancellations easier.