Here is how the number of opening positions are calculated.
Starting with one side. Finding two positions out of 15 to put the chariots : 105 ; 2 out of 13 points to put the horses: 78 ;2 out of 11 points to put the cannons :55 ;2 out of 9 points to put the elephants :36 and then 2 out of 7 points to put the advisors :21 . There is only one way to put the pawns in the remaining five positions since they are all indistinguishable. The total : 105 x 78 x 55 x 36 x 21 = 340540200.
The total for the other side is the same and is independent of this side ( This is so because the pieces are all shuffled by one’s opponent to ensure randomization.)
The total is therefore 340540200 * 340540200= 115967627816040000
About 3 years ago, I posted something about Reveal Chess, a variant of Xiangqi ( Chinese chess ) where the pieces are all turned upside down and then randomly placed on 15 starting positions. The King( General ) is the only piece placed at the original position. Based on the Cantonese dialect, Reveal Chess is pronounced as Kitkay, which seems more pronounciation-friendly than the Putonghua version Jieqi ( at least for me. )
Without the same restrictions placed on Chess 960 ( where the king must be between two rooks and the bishops must start on squares of different colours), Kitkay has the amazing number of starting positions as large as 115967627816040000. There is a Chinee saying about such numbers: As numerous as grains of sand on River Ganges. To put this into perspective, it is about 142 x 960 x 960 x 960 x 960 x 960 or 960 to the power 5.7219.
The fun of Kitkay is that there is a high degree of chance involved without greatly sacrificing the element of skill.
Here is a demo which I posted 8 months ago on Kitkay:
https://www.chess.com/forum/view/chess960-chess-variants/interesting-demo-of-reveal-chess