[Request] how to rank hands for poker if you used 6 cards instead of 5

Let's calculate the odds that you have a given hand. Here are the assumptions, 52 card deck (no wilds) and you are dealt exactly 6 cards. Also straights and flushes require all cards to be in it; e.g., not calculating five card straight flush/straight as a new separate hand.

To simplify the math, I'm calculating the odds you have this hand or better (in almost all cases). For example, when calculating the odds of say a flush, I'm finding the odds that all 6 cards have the same suit -- not necessarily the odds that all 6 cards have the same suit and also don't make a higher hand like a straight flush. Similarly, if you are dealt 2/2/3/3/3/K that counts as a high card, pair, two pair, three-of-a-kind, full house.

Note there are nine distinct straights with the high card being A,K,Q,J,10,9,8,7,6 (that is A-K-Q-J-10-9 to 6-5-4-3-2-A).

Please familiarize yourself with the notion of combination in math; e.g., choosing six distinct cards from a 52 card deck has nCr(52, 6)=20358520=52!/(6!*46!) different combinations of cards, which we use to find the odds of a hand. (You had 52*51*50*49*48*47=14658134400 ways of selecting the cards, but 6*5*4*3*2*1=720 possible orderings of these cards that you divide by when you don't care about order - in poker we treat the two-card hand A♠ K♥ is the same as K♥ A♠).

  1. Six-card Straight Flush - there are 36 straight flushes (9 straights x 4 suits), so 1 in 565514.44 hands is a six-card straight flush.
  2. Four of a kind and a pair - There are 13 choices for the rank of the four of a kind (one of each suit), 12 choices for rank of pair (can't be same rank as the 4 of a kind); and for the pair you also have 6 suit combinations (♠♥, ♠♦, ♠♣, ♥♦, ♥♣, ♦♣), hence 13*12*6=936 of these combinations, so 1 in 21750.55 hands
  3. Three-of-a-kind + three-of-a-kind. There are 13 choices for the rank of the first three of a kind, 4 choices for the suit left out of the first 3 of a kind, 12 choices for rank of the second three of a kind, 4 choices for suit left out of the second 3 of a kind; hence 13*\412\4=2496 combinations, so 1 in 8156.46 hands
  4. Six-card flush. You have 4 suits times 13 choose 6 (nCr(13,6)=1716) ways of selecting six cards from a rank, hence 4*1716=6864 of these combinations, so 1 in 2965.98 hands.
  5. Four of a kind. There are 13 choices for the rank of the four of a kind; then you have 48 cards left in the deck and can choose any two for the last two cards (48 choose 2 = 48*47/2=1128 choices), hence 13*48*47/2=14664 of these combinations, so 1 in 1388.33 hands
  6. Six-card straight. There are 9 straights times 46 ways of choosing the suits (4 choices for each card's suit), hence 36864 of these combinations, so 1 in 552.26 hands
  7. Full House (Three-of-a-kind + pair). There are 13 choices for the rank of the three of a kind, 4 choices of suit left out of three-of-a-kind, 12 choices for rank of the pair, 6 suit combinations for the pair, plus 1 free card (47 choices). Hence 13*4*12*6*47=175968 combinations, or 1 in 115.694 hands.
  8. Three pair (see note). There are 13 choices of rank of first pair, 12 choices of rank for second pair, 11 choices of rank of third pair, and each pair has 6 choices of suit, so 13*12*11*6*6*6=370656 combinations, or 1 in 54.93 hands
  9. Three of kind. 13 choices of rank of 3 of kind, 4 choices of rank left out in three-of-a-kind, three free cards (49 choose 3 = 49*48*47/6), hence 958048 combinations or 1 in 21.25 hands.
  10. Two pair. 13 choices for rank of first pair x 6 suit combinations, 12 choices for second pair x 6 suit combinations, plus two free cards (48 choose 2 = 48*47/2), which is 6334848 combinations or 1 in 3.21 hands
  11. One pair. 13 choices for rank of pair and 6 suit combinations, and four free cards (50 choose 4 = 50*49*48*47/24), hence 17963400 combinations or 1 in 1.13 hands.
  12. High card. You always have at least this, so 1 in 1 hands.

To summarize:

Hand Combinations Odds
Straight Flush 36 1 in 565514.44
4-kind + Pair 936 1 in 21750.55
3-kind + Three-of-kind 2496 1 in 8156.46
Flush 6864 1 in 2965.98
4-kind 14664 1 in 1388.33
Straight 36864 1 in 552.26
Full House (3-kind + pair) 175968 1 in 115.694
Three Pairs (pair+pair+pair) 370656 1 in 54.93
3-kind 958048 1 in 21.25
Two Pairs (pair+pair) 6334848 1 in 3.21 hands
Pair 17963400 1 in 1.13
High Card 20358520 1 in 1

Please note in step 8 (three pair), I assume the three pairs have different ranks. Thus something like A♠A♥K♠K♦A♦A♣ which is technically both (4-kind+pair) as well as as three pairs, I didn't include as being three pairs (as it is natural to exclude this possibility). Similarly for two pairs I don't include the possibility of 4-kind (like A♠A♥K♠Q♦A♦A♣) being considered a two pair.

/r/theydidthemath Thread