Why is $\frac{ r(r+1)\cdots(z+r-1)}{z!} = \binom{z+r-1}{z}$

Question

Sorry for this simple question, I am having a bit of trouble understanding how we can know immediately see that the LHS can be written as a binomial coeeficient

Answer 1

Just multiplying the top and bottom of the LHS by $1\cdot 2 \cdot 3\cdots (r-1)$ as follows:

$$ \begin{align} LHS &=\frac{1\cdot 2 \cdot 3\cdots (r-1)r(r+1)\cdots(z+r-1)}{1\cdot 2 \cdot 3\cdots (r-1)z!}\\ &= \frac{(z+r-1)!}{z!(r-1)!} \\ &=\binom{z+r-1}{z}. \end{align}$$