$\prod$ and factorial

Question

$\prod_{i=0}^{j-1}(j-i+1)$ is $=$ to $(j+1)!$ or $\le$? I think it is $=$, but if it's not, please put an explanation.

Answer 1

$$ \prod_{i=0}^{j-1}(j-i+1)=(j+1)\cdot j\cdot (j-1)\cdots 3\cdot 2=(j+1)! $$

Answer 2

\begin{align} \prod_{i=0}^{j-1} (j-i+1) & = (j+1)\cdot j\cdot (j-1)\cdots(j-(j-2)+1)\cdot (j-(j-1)+1)\\ & = 2\cdots (j+1)=(j+1)! \end{align}

Answer 3

Set $j-i+1=t$

$i=0\implies t=j+1$

$i=j-1\implies t=?$

$$\prod_{i=0}^{j-1}(j-i+1)=\prod_{t=j+1}^2t=\prod_{t=2}^{j+1}t=?$$