Value of a double indexed product

Question

Let $n$ be a positive integer. While trying to calculate the product $\prod_{1\leq i\leq j \leq n}^{} ij $, my approach was to write : $$\prod_{1\leq i\leq j \leq n}^{} ij=\prod_{j=1}^{n} \prod_{i=1}^{j} ij$$ But I am not sure if this is correct or not. Can someone please enlighten me?

Answer 1

Yes, this is valid. You are just traversing the same set of $(i,j)$ that constitute your index set in a different order. But you should note that

• This would be true for any $a_{i,j}$, not just $a_{i,j} = i \cdot j$.
• You couldn't automatically do this if your index set was infinite.
• As @CameronWilliams' comment points out, while it is valid it may or may not be useful.