Calculate your expected winnings (or losses) if you throw away all but the J♦ and K♣. This means you get rid of 4♦, 5♠, 6♠. You might get a pair, 2-pair, 3-of-a-kind, 4-of-a-kind, full house, straight, or nothing. We will let R be the set of cards {2,3,7,8,9,10,Q,A} and S be the set of {4,5,6} that don't include 4♦, 5♠, 6♠.

• pair of kings or pair of jacks (this is the same for either one so we just think of how we do the jacks and multiply by 2): one other J and two other cards that don't make a pair.

  1. the two cards are from R, or

  2. one card is from R and one from S, or

  3. both cards are from S

• pair of queens or aces (this is the same for either one so we just consider how we do the queens and multiply by 2): this includes 2 queens, J♦ and K♣, and one card from the other 37 cards

• two pair incl jacks: one pair of jacks (incl J♦), one pair from R, last card is the K♣.

• two pair incl kings: one pair of kings (incl K♣), one pair from R, last card is J♦.

• two pair incl jacks: one pair of jacks (incl J♦) one pair from S, last card is K♣.

• two pair incl kings: one pair of kings (incl K♣), one pair from S, last card is J♦.

• two pair; one of jacks and one of kings: one pair of jacks (incl J♦), one pair of kings (incl K♣) and one card from the remaining 41 cards

• 3-of-a-kind jacks or 3-of-a-kind kings (this is the same for either one so we decide how to do the jacks and multiply by 2): the J♦ and two more jacks, the K♣, and one more card from the remaining 41.

• 3-of-a-kind from R plus the J♦ and K♣.

• 3-of-a-kind from S plus the J♦ and K♣.

• 4-of-a-kind kacks plus the K♣.

• 4-of-a-kind kings plus the J♦.

• full house with 3 jacks and 2 kings

• full house with 2 jacks and 3 kings

• straight K high: one of each 9,10,Q

• straight A high: one of each 10,Q,A

• nothing: can be found by taking C(47,3) minus all previous hands OR

  1. three different cards from R (remove all possible straights)

  2. two different cards from R, one from S

  3. one card from R, two different cards from S

  4. three different cards from S

  5. pair of 4s or pair of 5s or pair of 6s and one other card from the remaining 38

  6. pair from {2,3,7,8,9,10} and one other card from the remaining 37