Dividing Factorial formula from book

Question

I'm reading a book Discrete Mathmatics with Cryptographic Applications and it claims that $\frac{n!}{k!} = (k+1)(k+2)...n$ for $k<n$. But a very simple example where $n = 3$ and $k = 2, \frac{(1)(2)(3)}{(1)(2)} = 3 \neq (2+1)(2+2)(2+3)$ What am i misunderstanding here?

Answer 1

The expression $(k+1)(k+2)\ldots n$ is the product of the numbers $k+1$, $k+2$, …, $n$. When $n=3$ and $k=2$ , there is only one such number: $2+1(=3)$. So, the product is $3$.

If $n=4$ and $k=2$, there are two such numbers: $2+1(=3)$ and $2+2(=4)$. And, indeed,$$\frac{4!}{2!}=12=3\times4.$$

Answer 2

It goes up to $n$, not $k+n$. In this case the product has only one factor namely $k+1 = 3$.