**Total Combinations**- In Holdem, you are a supplied with 7 cards, 2 in your hand and 5 community cards. With 7 cards, there are- 52 choose 7
- =52*51*50*49*48*47*46/7/6/5/4/3/2
- =133784560 combinations
- So there are 133784560 possible combinations one can get using 7 cards.

**Royal Flush**- Of the 133784560 combinations, there are- 4 Royal Flush hands that requires 5 cards and 2 remaining unused cards
- 4 * 47 choose 2
- =4324
- So there are 4324 possible cominiations to get a Royal Flush
- The chances of you getting a Royal Flush is 4324/133784560 = 0.00003232062

**Straight Flush**- To prevent overlaps in counting, we can break into cases**Case A**: seven cards straight flush- By counting with finger, there are Ace-7 to 8-Ace, a total of 8 combinations in each suit
**Case B**: exactly six cards straight flush- There are 9 combinations times the one dangling card that cannot be adjacent
- The outside straights where the hand has an Ace-6 straight, the dangling card cannot be a 7. When the straight hand is a 9-Ace, the dangling card cannot be an 8
- So the calculation when the straight hand contains an Ace is 2*45=90
- The inside straights means the dangling card cannot be two cards.
- When the straight hand does not contian an Ace, is 7*44=308
**Case C**: exactly five cards straight flush- There are 10 combinations times the two dangling card that cannot be adjacent
- Using the same idea equates to 2*(46 choose 2)+8*(45 choose 2)=9990
- In total, we also need to multiply by the four suits, we get
- (8+90+308+9990) * 4 = 41584
- So there are 41584 combinations of straight flushes.
- The odds of getting a straight flush is 41584/133784560= 0.00031082809
- One thing to note is that all Royal Flushes are Straight Flushes. So that calculation includes Royal Flushes

**Four of a kind**There are, by counting with your fingers, 13 four of kind hand using four cards. So in total, there are- =13 * 48 choose 3
- =224848
- Odds: 224848/133784560=0.00168067226

**Full House**- Breaking this into cases**Case A:**2 triplets and one single- (4 choose 3)*13*(4 choose 3) *12*11*4/2=54912
**Case B:**1 triplet, and 2 pairs- (4 choose 3)*13*(4 choose 2)*12*(4 choose 2)*11/2 = 123552
**Case C:**1 triplet, 1 pair and 2 singles- (4 choose 3)*13*(4 choose 2)*12*11*4*10*4/2=3294720
- In total,
- 54912+123552+3294720=3473184
- Odds: 3473184/133784560 = 0.0259610227

**Flush**- There are 13 choose 5 = 1287 combinations to get a five card flush in each suit.- Total combinations of flushes in all four suits using five cards is
- 1287 * 4 = 5148
- One again, we need to break this into three cases
- Case A: all seven cards in the same suit
- =4 *(13 choose 7)=6864
- Case B: six cards in the same suit and one card in a different suit
- 4 *(13 choose 6) *39=267696
- Case C: five cards in the same suit and two cards in a different suit
- 4 *(13 choose 5)* (39 choose 2)=3814668
- In total
- 6864+267696+3814668=4089228
- Odds: 3950804/133784560 = 0.02953109088
- Do note that the above calculation include straight flush

**Straight**- With straights, we have to remove flushes- 7 distinct cards generate 4*4*4*4*4*4*4=16384 unique cards
- Of the 16384 uniques cards, there are exactly 4 seven card flushes, 4*3*7 six card flushes and 4*3*3*(7 choose 2) five card flushes
- 4+84+756
- =844
- To remove flushes from any 7 distinct cards would be 4*4*4*4*4*4*4-844=15540
- With 6 distinct cards where the seventh card is a matching pair, there would be 4 six card flushes and 4*3*6=72 five card flushes should the mathching pair not exist
- We also need to make sure that the match pair doesn't create a flush as well. So the matching pair will create a flush 4*3*5=60 times
- 4+72+60=136
- To remove flushes from any 6 distinct cards would be 4*4*4*4*4*4-136=3960
- We need to remove flushes as we calculate the straights
**Case A1:**7 card straight- 8*(15540)=124320
**Case A2:**6 card straight with no pairs- 2*((15540)*6)+7*((15540)*5)=730380
**Case A3:**5 card straight with no pairs- 2*((15540)(7 choose 2))+8*(15540)*(6 choose 2))=2517480
**Case B1:**6 card straight with a pair matching the straight. We have to divide by 2 because the pairs generate overlap- 2*((3960*(18)/2))+7*((3960*(18)/2))=320760
**Case B2:**5 card straight with exactly one pair- 2*((3960*(18 choose 1)/2*(7 choose 1)))+8*((3960*(18 choose 1)/2*(6 choose 1)))=2209680
**Case C1:**5 card straight with a triple- 10*((4*4*4*4*4*(5 choose 1)*(3 choose 2)/3))=51200
- Of 51200, there are 10*4*(5 choose 1)*(3 choose 2)*10=600 flushes
- 51200-600=50600 straights with triples
**Case C2:**5 card straight with two different rank cards matching the straight as pairs- 10*(4*4*4*4*4(5 choose 2)*3*3/4)=230400
- Of the 230400, there are 10*4*(5 choose 2)*3*3=3600
- 230400-3600=226800
- In total,
- 124320+730380+2517480+320760+2209680+226800+50600=6180020
- odds: 6180020/133784560 = 0.04581560084

**Three of a kind**- using the same calculations ideas...- 6 461 620
- Odds: 6461620/133784560 = 0.04829869754

**Two Pairs**- Once again, same idea- 31 433 400
- Odds: 31433400/133784560 = 0.23495536405

**A Pair**- ....- 58 627 800
- Odds: 58627800/133784560 = 0.4382254574

So now you know the poker odds of each hand. But knowing this, although is nice, is not really helpful. So, my next topic will be the helpful basics. If I have not written, it is coming.