A special deck has 5 suits of 13 cards each, making a total of 65 cards.
We are dealing a 6-card hand from this deck. In how many ways one can pick a "well-suited" hand (defined below)?
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}.
My attempt: Since there are 5 suits and we pick 6 cards, there will be one suit of which there will be 2 cards. We will pick one card from all other suits.
$$\binom{5}{1} \left (\binom{13}{2}-12 \right)11\cdot 10 \cdot 9 \cdot 8 $$
Explanation:
- There are 12 ways to get consecutive numbers from the suit e.g. 1&2, 2&3 etc, hence subtracting 12 from $$\binom{13}{2}$$.
- If we pick 2 & 4 from deck A, then we can't pick these numbers/ranks from any other deck (since pairs are not allowed), hence from the next deck, only 11 options remain, then 10 options remain, and so on.
I am not very sure if what I am doing is correct.