Finding the error in my pearl puzzle solution

[Lauren1234]

Homework Statement

both the type and the suit of the card are important. Each pack again consists of eight cards; the Jack and Queen of Hearts and Diamonds, and the King and Ace of Spades and Clubs. A ‘snap’ occurs when two cards that are either of the same type OR suit appear together. Find the total number of possible orderings of the cards which result in no snaps. (You may first wish to assume that the order of the cards in the first pack is fixed.)

Relevant Equations

Below

This is my solution however I feel like the number is far too big can anyone see what I’ve done wrong

 

[mitochan]

Interesting problem. I have not solved it thoruogh but let me tell you my idea.

We can make use of necklace made of eight pearls whose color is Red or Black to express no snap order of cards.

Let us show the no snap order of pearl colors like

(1)BRBRBRBR : alternative or no color consecutives at all

No more than three consecutive same colors, like BBB, is allowed.

Case of each two BB and RR pairs in ring

(2)BBRRBBRR

Cases of each one BB and RR pair in ring

(3)BBRRBRBR : No pearls between BB and RR and

(4)BBRBRRBR: Two pearls between BB and RR

To these color cases we can think of the cases of JQ and HD for Red and KA and SC for Black.