7-Card Six Plus Hand Frequencies
Since the game the players were discussing was Six Plus Hold-Em, we need to look at the frequencies of 7-card hands rather than, or in addition to, 5-card frequencies.
Here are the detailed derivations that my daughter and I slogged through (undoubtedly there are easier or more condensed ways to do this).
Category 1: Royal Flush
4*C((36-5),2) = 1,860
Category 2: Straight Flush (excluding royal flush)
4*5*C((36-6),2) = 8,700
Category 3: Four of a Kind
9*C((36-4),3) = 44,640
Category 4: Flush (not royal or straight flush)
Seven cards are in same suit = 4*C(9,7) = 144
Six cards are in same suit = 4*C(9,6)*C(27,1) = 9,072
Five cards are in same suit = 4*C(9,5)*C(27,2) = 176,904
So we have (144+9072+176904) - (1860+8700) = 175,560
Category 5: Full House
Ranks are [3,3,1] = C(9,2)*C(4,3)*C(4,3)*C(7,1)*C(4,1) = 16,128
Ranks are [3,2,2] = C(9,3)*3*C(4,3)*C(4,2)*C(4,2) = 36,288
Ranks are [3,2,1,1] = C(9,2)*2*C(4,3)*C(4,2)*C(7,2)*C(4,1)*C(4,1) = 580,608
Total = 633,024
Category 6: Three of a Kind (no full house, flush, straight-flush, or royal flush)
Ranks need to be [3,1,1,1,1] = C(9,1)*C(4,3)*C(8,4)*(4^4) - 4*C(9,5)*5*C(3,2) = 637,560
Category 7: Straight (no three of a kind, flush, straight-flush, or royal flush)
Ranks are [2,2,1,1,1] = 6*C(5,2)*C(4,2)*C(4,2)*C(4,1)*C(4,1)*C(4,1) = 138,240
Ranks are [2,1,1,1,1,1] & pair is part of straight:
Ace high straight = 5*C(4,2)*(4^4)*C(4,1)*4 = 122,880
King-Nine high straight = 5*5*C(4,2)*(4^4)*C(3,1)*4 = 460,800
Ranks are [2,1,1,1,1,1] & pair is not part of straight:
Ace high straight = (4^5)*C(4,1)*C(4,2) = 24,576
King-Nine high straight = 5*(4^5)*C(3,1)*C(4,2) = 92,160
(Subtotal of ranks are [2,1,1,1,1,1] = 700,416 (we will use this below))
Ranks are [1,1,1,1,1,1,1]:
Ace high straight = (4^5)*C(4,2)*(4^2) = 98,304
King-Nine high straight = 5*(4^5)*C(3,2)*(4^2) = 245,760
(Subtotal of ranks are [1,1,1,1,1,1,1] = 344,064)
Subtotal of all straights = 1,182,720
Now we need to subtract off straights that are also flushes:
Ranks are [2,2,1,1,1] = 4*6*C(5,2)*C(3,1)*C(3,1) = 2,160
(Subtotal of straights & flushes with ranks [2,2,1,1,1] = 2,160)
Ranks are [2,1,1,1,1,1] with pair part of straight:
Ace high straight with 6 suited cards, 5 in straight = 4*5*3*4 = 240
Ace high straight with 5 suited cards, 5 in straight = 4*5*3*4*3 = 720
Ace high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*3*4*3*4 = 2,880
Ace high straight with 5 suited cards, 4 in straight, pair not in suit = 4*5*C(3,2)*4 = 240
King-Nine high straight with 6 suited cards, 5 in straight = 4*5*5*3*3 = 900
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*5*3*3*3 = 2,700
King-Nine high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*5*3*4*3*3 = 10,800
King-Nine high straight with 5 suited cards, 4 in straight, pair not in suit = 4*5*5*C(3,2)*3 = 900
Ranks are [2,1,1,1,1,1] with pair not part of straight:
Ace high straight with 6 suited cards, 5 in straight = 4*4*1*3 = 48
Ace high straight with 5 suited cards, 5 in straight = 4*4*C(3,2) = 48
Ace high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*3*4*1*3 = 720
Ace high straight with 5 suited cards, 4 in straight, pair not in suit = 0 (impossible)
King-Nine high straight with 6 suited cards, 5 in straight = 4*5*3*1*3 = 180
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*3*C(3,2) = 180
King-Nine high straight with 5 suited cards, 4 in straight, pair in suit = 4*5*5*3*3*1*3 = 2,700
King-Nine high straight with 5 suited cards, 4 in straight, pair not in suit = 0 (impossible)
(Subtotal of straights & flushes with ranks [2,1,1,1,1,1] = 23,256)
Ranks are [1,1,1,1,1,1,1]:
Ace high straight with 7 suited cards, 5 in straight = 4*C(4,2) = 24
Ace high straight with 6 suited cards, 5 in straight = 4*C(4,2)*2*1*3 = 144
Ace high straight with 6 suited cards, 4 in straight = 4*5*3*C(4,2) = 360
Ace high straight with 5 suited cards, 5 in straight = 4*C(4,2)*3*3 = 216
Ace high straight with 5 suited cards, 4 in straight = 4*5*3*C(4,2)*2*1*3 = 2,160
Ace high straight with 5 suited cards, 3 in straight = 4*C(5,2)*3*3*C(4,2) = 2,160
King-Nine high straight with 7 suited cards, 5 in straight = 4*5*C(3,2) = 60
King-Nine high straight with 6 suited cards, 5 in straight = 4*5*C(3,2)*2*1*3 = 360
King-Nine high straight with 6 suited cards, 4 in straight = 4*5*5*3*C(3,2) = 900
King-Nine high straight with 5 suited cards, 5 in straight = 4*5*C(3,2)*3*3 = 540
King-Nine high straight with 5 suited cards, 4 in straight = 4*5*5*3*C(3,2)*2*1*3 = 5,400
King-Nine high straight with 5 suited cards, 3 in straight = 4*5*C(5,2)*3*3*C(3,2) = 5,400
(Subtotal of straights & flushes with ranks [1,1,1,1,1,1,1] = 17,724)
Putting it all together, we have:
= 1,182,720 - (2,160+23,256+17,724)
= 1,139,580
Category 8: Two Pair (no full house, flush or straight)
Ranks are [2,2,2,1] = C(9,3)*C(4,2)*C(4,2)*C(4,2)*C(6,1)*C(4,1) = 435,456
Ranks are [2,2,1,1,1] = C(9,2)*C(4,2)*C(4,2)*C(7,3)*C(4,1)*C(4,1)*C(4,1) = 2,903,040
Now subtract off flushes:
Ranks are [2,2,1,1,1] = 4*C(9,5)*C(5,2)*C(3,1)*C(3,1) = 45,360
Now subtract off straights:
Ranks are [2,2,1,1,1] = 138,240 (from above)
Now add back in all hands with ranks [2,2,1,1,1] that are both straights and flushes:
We know this is 2,160 (from above)
Putting it all together, we have:
= (435,456 + 2,903,040) - (45,360 + 138,240) + 2,160
= 3,157,056
Category 9: One Pair (no straight or flush)
Ranks need to be [2,1,1,1,1,1] = C(9,1)*C(4,2)*C(8,5)*(4^5) = 3,096,576
Now subtract off flushes with ranks [2,1,1,1,1,1]:
Pair is not part of flush = 4*C(9,5)*C(4,1)*C(3,2) = 6,048
Pair is part of flush, other card is suited = 4*C(9,6)*6*C(3,1) = 6,048
Pair is part of flush, other card is not suited = 4*C(9,5)*5*C(3,1)*C(4,1)*C(3,1) = 90,720
(Subtotal of flushes with ranks [2,1,1,1,1,1] is 102,816)
Now subtract off straights with ranks [2,1,1,1,1,1]:
We know this is 700,416 (from above).
Now add back in hands with ranks [2,1,1,1,1,1,] that are both straights and flushes:
We know this is 23,256 (from above).
Putting it all together, we have:
= 3,096,576 - (102,816 + 700,416) + 23,256
= 2,316,600
Category 10: High Card (no straight or flush)
Ranks need to be [1,1,1,1,1,1,1] = C(9,7)*(4^7) = 589,824
Now subtract off flushes with ranks [1,1,1,1,1,1,1]:
Seven cards in suit = 4*C(9,7) = 144
Six cards in suit = 4*C(9,6)*3*3 = 3,024
Five cards in suit = 4*C(9,5)*C(4,2)*3*3 = 27,216
(Subtotal of flushes with ranks [1,1,1,1,1,1,1] is 30,384)
Now subtract off straights with ranks [1,1,1,1,1,1,1]:
Ace high straight = (4^5)*C(4,2)*(4^2) = 98,304
King-Nine high straight = 5*(4^5)*C(3,2)*(4^2) = 245,760
(Subtotal of straights with ranks [1,1,1,1,1,1,1] is 344,064)
Now add back hands with ranks [1,1,1,1,1,1,1] that are both straights and flushes:
We know this is 17,724 (from above)
Putting it all together, we have:
= 589,824 - (30,384 + 344,064) + 17,724
= 233,100
