We have two constraints to arrange the standard $52$ playing cards:
I. No two adjacent cards have the same suits.
II. An ace does not have an adjacent ace or two. A two does not have an adjacent ace, two, or three. A three does not have an adjacent two, three, or four. A four does not have an adjacent three, four, or five. And so on. A jack does not have an adjacent ten, jack, or queen. A queen does not have an adjacent jack, queen, or king. Finally, a king does not have an adjacent queen or king.
In how many ways can we arrange these $52$ playing cards where the two given constraints are satisfied simultaneously?
Honestly, I do not know how to solve it. I only tried and got the result: $\boxed{52!-[4(11 \times 10^{12}+2 \times 11^{12})]}$
$\boxed{=80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,754,892,572,986,232}$
Any help would be appreciated. THANKS!