Equivalent Capital Pi Notation Expressions So I was working on a probability question and then this expression came up.
When I consulted the answers, I struggled to understand exactly how I would get from one expression to the other myself.

Substituting a constant such as
let $n=5$ makes it a bit clearer how they got from one expression to the other.
But is there a simple, yet general explanation of these expressions.
I am hoping the community could give some insight into how I should have approached this problem and similar ones in future.


We obtain
  \begin{align*}
P_l(n)&=\prod_{k=1}^n\prod_{j=0}^{k-1}\frac{40-j}{52-j}\\
&=\prod_{\color{blue}{0\leq j<k\leq n}}\frac{40-j}{52-j}\tag{1}\\
&=\prod_{j=0}^{n-1}\prod_{k={j+1}}^n\frac{40-j}{52-j}\tag{2}\\
&=\prod_{j=0}^{n-1}\left(\frac{40-j}{52-j}\right)^{\color{blue}{n-j}}\tag{3}
\end{align*}

Comment:


*

*In (1) we rewrite the index range to see the relationship somewhat more conveniently.

*In (2) we change the order of the products and write lower and upper limits corresponding to the index range given in (1).

*In (3) we note that $\frac{40-j}{52-j}$ does not depend on the index $k$. This means we have a product of $n-j$ times the same factor.
