Simplification in expressions with factorials

It is possible to "cut short" a factorial expression by using the following equality: $n! = n \cdot (n - 1)!$

This allows us to simplify terms when factorials appear in fractions.

Example

For instance, calculating an expression like: $\frac{8!}{6! \cdot 3!}$ We must bear in mind that in the numerator $8! = 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 8 \cdot 7 \cdot 6!$ (we have stopped the development at $6!$ because it is the term that appears in the denominator and thus we will be able to simplify it. So that $\frac{8!}{6! \cdot 3!} = \frac{8 \cdot 7 \cdot 6!}{6! \cdot 3!} = \frac{8 \cdot 7}{3 \cdot 2} = \frac{28}{3}$

Example

Another example, calculating the value of: $\frac{14! \cdot 6!}{13! \cdot 7!}$ In this case we will develop the numerator and denominator in such a way as to have the biggest advantage in the simplification, $14! = 14 \cdot 13!$ and $7! = 7 \cdot 6!$ :

$\frac{14! \cdot 6!}{13! \cdot 7!} = \frac{14 \cdot 13! \cdot 6!}{13! \cdot 7 \cdot 6!} = \frac{14}{7} = 2$

The same method can be used for literal expressions (those in which letters appear instead of numbers): $\frac{x!}{(x - 1)!} = \frac{x \cdot (x - 1)!}{(x - 1)!} = x$

The example can be as complicated as you like, but the solution will always be simple: $\frac{(m - 2)! \cdot x!}{(x - 1)! \cdot m!} = \frac{(m - 2)! \cdot x \cdot (x - 1)!}{(x - 1)! m \cdot (m - 1)!} = \frac{(m - 2)! \cdot x}{m \cdot (m - 1) \cdot (m - 2)!} = \frac{x}{m \cdot (m - 1)}$