A peculiar deck of playing cards is like a regular deck, but has five suits: spades, hearts, diamonds, clubs, and clovers. There are 13 cards in each suit: Ace, 2, 3, …, 10, Jack, Queen, King. There are thus 5×13 = 65 cards in this deck. Consider dealing a 6-card hand from this deck.

A well-suited hand is one that contains at least one card of every suit, no more than one card of any kind (no pairs, etc.), and not all cards of consecutive kinds. An example of a well-suited hand is {2 of spades, 4 of hearts, 5 of diamonds, 8 of clubs, 10 of clubs, Jack of clovers}.

How many different 6-card hands can be dealt from this deck that are well-suited?

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I started off by using 5c1 to choose the one suit that gets a repeat, then multiply by 13c2 for the possible combinations of two cards to choose from that suit after that. I thought at this point that I could do 13^4 for the remaining cards, but then I realized that I have to avoid repeating anything (to avoid pairs). How can I do that? And how do I avoid getting all consecutive cards?