Simplification of factorials

Question

Just had a quick question about the simplification of the following factorial.

$$\frac{(n-1)!}{n!}=\frac{1}{n}$$

why does this simplify this way?

Answer 1

$\require{cancel}$ Because $n! := n(n-1)!$ (recursive definition of factorial)

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Intuitively, you can think of this:

$$\dfrac{(n-1)!}{n!} = \dfrac{\cancel{1 \cdot 2 \cdot 3 \cdots (n-1)}}{\cancel{1 \cdot 2 \cdot 3 \cdots (n-1)} \cdot n} = \dfrac 1 n$$

Answer 2

$\require{cancel}$This comes from the definition of a factorial.

Perhaps it is easier to see an example with a number.

Let $n=5$. Then we have:

$$\dfrac{(5-1)!}{5!}=\dfrac{4!}{5!}=\dfrac{4\times 3\times 2\times 1}{5\times 4\times 3\times 2\times 1}= \dfrac{\cancel{4\times 3\times 2\times 1}}{5\times \cancel{4\times 3\times 2\times 1}}=\dfrac{1}{5}=\dfrac{1}{n}$$