In a cards game, there are ten cards, $1$ to $10$. Two players, seated facing each other, randomly choose $5$ cards each. They arrange their cards in ascending order of the number on the cards as shown below.
https://i.stack.imgur.com/3sRqU.png
The difference between the corresponding cards is calculated such that the lower value is subtracted from the higher value.
In a random game, what is the probability that the sum of the differences is $24$?
The answer to this is $0$.
[I observed that in such an experiment, we obtain the sum of such differences as $25$ always].
How do I solve it?
Edit: To avoid any confusion, I want to make it clear that the image only implies that one player arranged his cards in ascending order from left to right and the other from right to left as observed in top view of the table from either player's end.
Hint: prove that no matter how you arrange the cards, the cards $1,2,3,4,5$ are opposite some higher-value card, while the cards $6,7,8,9,10$ are opposite some lower-value card. (Think about it one card at a time: what would it take for $5$ to end up opposite one of $1,2,3,4$, for example?)
If you prove this, then it follows that taking the sum of positive differences is equivalent to the calculation $$10+9+8+7+6-5-4-3-2-1,$$ just done in a different order, so it is guaranteed to yield $25$.