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PL/NL Texas Hold'em >> Micro Stakes

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derosnec
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Reged: 10/11/05
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Combinatorics (very very long)
      #10760165 - 06/12/07 12:42 PM

Chomp asked me to post about counting combos, so I have some time now.

For some (or most, I donít know) this post might be obvious and second nature. But maybe for a few it will be helpful. While reading this, keep in mind that I have an IQ of about 20 or 30 in the mornings and that I'm not a coach or anything, so if there are errors, please tear into me all you want. I like to learn.

Combinations 101

Itís easy to do. Take the remaining cards and multiply them by each other. Check Poker Stove to verify.

Example: Flop is A67. You have QQ.

How many ways can the opponent have AK? 3 aces remaining multiplied by 4 kings remaining = 12 ways he can have AK. So 48 ways for AT+.

How about 67? 3 * 3 = 9 ways.

What you will notice after doing this after while is youíll start to gain some insights. For example, there are a lot of straight draws on many flops. On A67, there are 48 ways he can have an OESD (16 ways of 98, 16 ways of 54, 16 ways of 85) 64 ways he can have a gutshot (16 ways of T9, 16 ways of 43, 16 ways of 84, and 16 ways of 95). Thatís 112 ways he can have a straight draw (ok, 95, 84, and 85 arenít very realistic, but you get the point).

Letís try AA on a QJJr flop. You are the pfr and bet. He calls.

Are you generally ahead here?

Assume we put his pf range at 22+, A8+, K9+, Q9+, J9+, T8+, 98, 76.

The first thing youíll notice is the ways he can have top pair. AQ = 6 ways (because 2 aces left and 3 queens left); KQ = 12 ways; QT = 12 ways; Q9 = 12 ways. So thatís 42 ways he can have TP.

What about straight draws? 32 ways he can have an OESD; 8 ways he can have AT; 8 ways for AK; 16 ways he have T8. And 16 ways he can have K9. 16 ways for 98. Thatís 96 ways he can have a straight draw. Impressive, huh?

And trips or better? 4 ways he can have AJ; 8 ways he can have KJ; 3 ways he can have QQ; 6 ways for QJ; 1 way JJ; 8 ways for JT; 8 ways for J9. 38 ways he can have trips or better. Thatís almost as many ways as he can have top pair.

The other hands: weíll assume he folds 22-88/A8/A9 to the flop bet. So, that leaves 6 ways for KK, 6 ways for TT, and 6 ways for 99. 18 ways total.

Add it all up and we get:

Weíre ahead of 156 combos.

Weíre behind 38 combos.

Range versus Combinations

The example above is a bit more revealing (to me at least) of the reality of opponentís distribution of hands than merely just saying he has X range.

If you are only using ranges, you are Ė to use an analogy - looking at a piece of bread, while those who use combinations are looking at that bread with a microscope and seeing all that constitutes it (maybe thatís a dumb analogy).

Reads versus Combinations

You might argue, ďBut I have my reads, man. I donít need to count no stinking combinations.Ē

Reads, in my opinion, have severe limitations and are not as good guideposts in assessing the strength of our opponentís hand postflop as we think they are.

The reasons why might be obvious but Iíll discuss them anyway.

1)Sample size: Say you have 300 hands on an opponent (which is a lot of hands against a micro opponent since we are not playing the same people often, as opposed to MSNL/HSNL). Assume that opponent has a VPIP of 20 and WTSD of 25. That means Ė at best Ė you saw 15 showdowns where he put money in preflop (not checked his BB). So you have direct hand evidence of 15 hands. What do 15 hands tell you? Very little. See the next sections for why.

2)Preflop dynamics. Sometimes an opponent is the pf raiser, sometimes he is the caller. This affects how he plays his hands postflop. In other words, he might play TT different as the pf caller on a Qh6s7s flop than he would as the pf raiser.

3)Number of opponents posftlop: How an opponent plays top pair, a draw, or a monster multiway might differ how he plays it heads up.

4)Flop texture: There are roughly 17,000 flop combinations heads up (I think). Our reads are based on a tiny % of those flops that opponent has seen. Take opponent having a set for example. Will he play it the same on all flop combinations? Our read on how an opponent plays a set will probably come from only one or two flop combinations observed.

5)Position: How an opponent plays a hand out of position, on the button, or in good/bad relative position will likely affect how he plays his hands postflop. So, he might play a set different OOP than he would on the button or in good versus bad relative position.

6)Steal situations: An opponent might call more/be more aggressive in these situations than in non-steal hands.

7)Your image: This is very important from my experience. When I tightened up over my last 100k hands (my vpip dropped to about 14 or 15), I felt like I was getting coolered a lot. Maybe I was. But what was probably happening is opponents were playing their hands differently against me than they were against other opponents. So, while an opponent would play TPGK fast against another player, he was playing it slow and cautiously against me, but playing his sets fast against me.

General Micro Plays/Theorems versus Combinations

A lot of players at the micros make the same plays. They minraise with sets, slowplay the flopped nut flush, donít c/r bluff the river, put you on AK, donít fold full houses (Zeebo Theorem), raise turn with better than top pair (Baluga Theorem), etc.

So, for these situations, combo counting is of less value.

The Not-So-Obvious Value of Combinatorics

So, assuming youíve made it this far in the thread, you might be thinking, ďNo way in hell am I doing all this counting at the table. I donít have a time bank of 2 hours per hand.Ē

You want to review your hands in PT and apply combo counting to see if you made good decisions. After you do that for a while, counting will become much quicker and intuitive because you will have seen so many similar scenarios. So, counting top pair combos on the flop, for example, will require little thought.

Now, given this practice, you will start to gain quite a few insights. When you begin to apply these insights, you will become a much tougher player. What will begin to happen is that your game will start to lean towards a Game Theory-ish Optimal strategy.

True, you are exploiting your opponents based on the ways they can have a hand, but by doing that you start to make your plays less based on your own hand and more against the distribution of your opponentsí hands, which makes your opponent's task of exploiting you very very difficult.

An example:

So, after spending way too much time analyzing PT hands, I noticed that I was doing a lot of this:

tag UTG/MP raises pf. I call with a small pp for set value.

Heads up, the flop comes A62r. I check, he c-bets. I fold having not made my set.

But after playing around with combo stuff, I determined that they do not have a pair of aces or better on that flop the majority of the time. So I started to check-raise here or raise from LP (doing it in a way so that it is profitable based on pot size).

Now I was no longer playing my hand. Of course, if I do this 5 times in a row against him he will likely adjust, but I can flop sets/two pair/etc too.

And on turns and rivers, your play will naturally become much tougher. If I end up on a river, and given opponentís play so far I assess that there are 70 ways he can have a better hand and 40 ways he can have a worse hand, but only 24 ways he can have a monster, I can play aggressive enough to cause him to fold more often than he should.

Also, by playing with Poker Stove, another insight you will likely gain is how few legitimate flush draws an opponent can have compared to other hands. For some reason our brain spots flush draws right away, despite the reality that they are often a tiny part of the ways he can have a hand.

Weighting

Just because someone has a range of XYZ, doesnít mean X is as likely as Z. So, in Poker Stove, you go through a hand and reduce the ways he can have a hand so you can weight them appropriately. So, if KTo is in his range, but rarely so given pf and flop action, Iíll reduce it from say 12 combos to 2 combos by only selecting KcTd and KhTc, for example.

Warning

At 10nl or lower, where your opponents are rarely engaged in the process known as "thought", be very careful about ever bluffing. At 25nl, there are a few tags, and at 50nl, there are alot of tags who do think. So, keep that in mind.

To see combo counting in action by one of th ebest, look up Bobbo Fittos' posts. This a good one (his analysis is towards the end of the thread): Link

Edited by derosnec (06/12/07 01:10 PM)


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bozzer
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Re: Combinatorics (very very long) [Re: derosnec]
      #10760212 - 06/12/07 12:46 PM

w00t!

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matrix
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Re: Combinatorics (very very long) [Re: bozzer]
      #10760280 - 06/12/07 12:51 PM

derosnec - good stuff.

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tms
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Re: Combinatorics (very very long) [Re: matrix]
      #10760315 - 06/12/07 12:54 PM

wow awesome stuff, going to study/work on it right now

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Bonesy
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Re: Combinatorics (very very long) [Re: derosnec]
      #10760395 - 06/12/07 01:02 PM

Finally someone explained the combo platter. Thanks a lot!

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poisonxfree
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Reged: 10/26/05
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Re: Combinatorics (very very long) [Re: Bonesy]
      #10760425 - 06/12/07 01:05 PM

Awesome. Bookmarking this for next week when I can actually start playing again.

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Waingro
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Reged: 11/23/05
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Re: Combinatorics (very very long) [Re: derosnec]
      #10760430 - 06/12/07 01:05 PM

Lol I feel so retarded after reading posts like this. Clear, easy to read and very helpful. I am more in the "LOLZ I have aces I push" camp, but I will try to use this approach sometimes as well. Especially when reviewing a hand in pt.

Sticky plz obv.


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Happy Birthday Fiksdal
banned


Reged: 08/22/06
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Re: Combinatorics (very very long) [Re: Waingro]
      #10761007 - 06/12/07 02:48 PM

tldr

I certainly will though, during one of the boring classes this week. looking forward to it


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Chomp
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Reged: 05/22/06
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Re: Combinatorics (very very long) [Re: Fiksdal]
      #10761067 - 06/12/07 02:54 PM

Jesus, about time.

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Bowlboy
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Re: Combinatorics (very very long) [Re: Fiksdal]
      #10761213 - 06/12/07 03:06 PM

Nice post. I gave this stuff some good thought awhile ago by pulling out a deck and figuring out the number of combinations for non-paired hands though I have to admit I have'nt practiced counting them after sessions as you suggested and that is a great idea.

I just wanted to add for beginner combo counters that for pocket pairs the total possible combinations is 6 ways. I dont know what the calculation is, I believe it's a permutation though not sure, I'm useless at math. So if you want to know how many combinations are possible for AA it's 6. If there is an Ace on the flop then there are 3 aces left and that also means there are 3 combos. With 2aces out there is only obviously 1 combo. So far pocket pairs just remember 6,3,1.


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bluffbetter
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Re: Combinatorics (very very long) [Re: Bowlboy]
      #10762142 - 06/12/07 05:26 PM

The 6 ways is because there are 6 combinations of suits for each pair. e.g.

A A ..........A A
A A ..........A A
A A ..........A A


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Nottom
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Re: Combinatorics (very very long) [Re: Bowlboy]
      #10763478 - 06/12/07 06:57 PM

Quote:

I just wanted to add for beginner combo counters that for pocket pairs the total possible combinations is 6 ways. I dont know what the calculation is, I believe it's a permutation though not sure, I'm useless at math.




[mathgeek]
Technically its a combination. A permutation is when order matters, but since A A = A A we then divide the number of permutations (4*3) by the number of ways we can arrange each one (in this case 2) to get the number of combinations (6)
[/mathgeek]


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Antinome
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Re: Combinatorics (very very long) [Re: derosnec]
      #10763764 - 06/12/07 07:19 PM

nice.

while combining this with equity calculations is implicit, some explicit examples would be cool.


Also the microscope metaphor- I tend to look at range->combinatorics->action as a series of filters that continuously narrow the range. I don't always do the combinatorics right, but in some of those really tricky spots in 3bet pots where the range starts narrow it is pretty easy and helps a lot.

Edited by Antinome (06/12/07 07:32 PM)


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cooker3
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Re: Combinatorics (very very long) [Re: derosnec]
      #10764754 - 06/12/07 08:36 PM

Great post.
This has been something I am thinking about more but too lazy too actually do it. It just goes to show how much I have to learn.
Sticky this plz


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Chomp
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Re: Combinatorics (very very long) [Re: cooker3]
      #10769729 - 06/13/07 07:31 AM

Derosnec, could you post a hand where you think you used this in real-time at the table?

Cheers. Great post.


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kazana
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Re: Combinatorics (very very long) [Re: Chomp]
      #10769791 - 06/13/07 07:45 AM

Love it. Thanks.

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Jouster777
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Re: Combinatorics (very very long) [Re: tms]
      #10769835 - 06/13/07 07:55 AM

Very nice derosnec! This stuff really has a very small barrier to understanding and if people just read your post a couple times it WILL become second nature.

The combinatorics should be the easy part. Correctly assessing the dynamics of the hand (villain's play, PF, texture, our image, etc.) and then translating that to a range with proper discounting is the hard part. After that the combos are automatic (or should be). I hope this post gets more people doing it because it will raise the level of discussion on the whole board. Once someone goes through the trouble of dissecting a hand with combos, the discussion can switch to assessing the assumptions that went into the calculation rather than staying at the level of "well I always push here".


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corsakh
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Re: Combinatorics (very very long) [Re: Bowlboy]
      #10769837 - 06/13/07 07:55 AM

Nice.

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Baintz
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Re: Combinatorics (very very long) [Re: derosnec]
      #10769850 - 06/13/07 07:58 AM

Great stuff, muchos thanks for doing this.

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bozzer
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Re: Combinatorics (very very long) [Re: Baintz]
      #10770678 - 06/13/07 10:12 AM

i read this last night, and it helped a lot - i didn't realise it was so easy to do.

I agree with Antimone that this is probably most useful in spots where ranges are very tightly defined - like in 3bet pots when an ace and a king flops or something.


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munkey
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Re: Combinatorics (very very long) [Re: bozzer]
      #10770971 - 06/13/07 10:42 AM

Good stuff

I just read this properly and never really considered counting the combos @ the table - a new concept to add.

Interesting conclusions on str8s and flushes. Some things I need to add to my pondering about flushes now I've been recently thinking about.

1. I'm playing more str8 draws as more deceptive than flushes ->> better implied odds esp when they come in. I also am terrible at spotting when they come in and actually have to write down the missing cards

2. How to play OOP[IP too FWIW] when a flush card hits the turn and we hold a made hand - may be able to semi-bluff it more as you say flush combos are much less likely and I suspected I'm losing value on flushy turns. I may also be misisng value by being too cautious with baby flushes.

[/ramble]

I'll have to do some practice -mainly in narrow rage 3bet scenarios like antinome alluded to before I can properly integrate this.


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Peter Harris
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Re: Combinatorics (very very long) [Re: derosnec]
      #10771191 - 06/13/07 11:09 AM

Quote:

Example: Flop is A67. You have QQ.

How many ways can the opponent have AK? 3 aces remaining multiplied by 4 kings remaining = 12 ways he can have AK. So 48 ways for AT+.





[nit]if we're holding QQ that takes out some combos, so it ain't 48[/nit]

goot post OP


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derosnec
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Re: Combinatorics (very very long) [Re: Chomp]
      #10771349 - 06/13/07 11:24 AM

Quote:

Derosnec, could you post a hand where you think you used this in real-time at the table?




Here are two examples (granted when counting at the table I did more estimates than precise counting):

Hand 1

PokerStars No-Limit Hold'em, $0.50 BB (5 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: 2+2 Forums)

MP ($183.05)
Button ($50.70)
SB ($50)
Hero ($48.25)
UTG ($30.30)

Preflop: Hero is BB with 2, 2.
1 fold, MP raises to $1.5, 2 folds, Hero calls $1.

Flop: ($3.25) 4, K, 3 (2 players)
Hero checks, MP bets $3, Hero raises to $10, MP folds.

Final Pot: $16.25

Here there is an argument of course for me folding pf, since I'm OOP against a tag (tough to extract). Anyways:

Flop comes high and dry. My hand is irrelevant here. I'm playing against his distribution of hands.

What hands can he call my raise with?

AK (12 ways)
KK (3 ways)
AA (6 ways)
33 (3 ways)
44 (3 ways)

So, that's 27 ways.

What hands will he fold?

QQ-55 (48 ways)
Probably KQ/KJ, because there are no draws really (and I'm tight), so that's 32 ways
AQ/AJ/AT/A9/QJ/JT and some SCs (since he is in a steal position), we'll estimate that at over 100 ways.

So, ignoring our hand, there are 180 ways he will fold and 27 ways he will call/raise. Once he calls/raises, I'm outta there.

Putting $10 into a $6 pot, I need him to fold a little more than 60% of the time to break even. Should I have bet less or more? Maybe. Still working on that part of my game.

Hand 2

PokerStars No-Limit Hold'em, $0.50 BB (5 handed) Hand History Converter Tool from FlopTurnRiver.com (Format: 2+2 Forums)

SB ($49.95)
BB ($61.60)
Hero ($61.70)
MP ($30.65)
Button ($49.25)

Preflop: Hero is UTG with A, K. MP posts a blind of $0.75.
Hero raises to $2.75, 3 folds, BB calls $2.25.

Flop: ($6.50) K, T, 7 (2 players)
BB checks, Hero bets $4.5, BB calls $4.50.

Turn: ($15.50) Q (2 players)
BB checks, Hero bets $9.5, BB calls $9.50.

River: ($34.50) Q (2 players)
BB bets $23, Hero folds.

Final Pot: $57.50



Opponent is somewhat loose

Again, first thing I do is count ways I'm beat:

1 way KK
3 ways TT
3 ways 77
6 ways KT

Ways I'm ahead

One pairs:
8 ways KQ
8 ways KJ
9 ways AT (weighted to about 1 since I doubt people counted with it)
6 ways QQ (weighted to about 2 since he probably would fold flop)

Straight draws:
12 ways AQ
16 ways QJ
16 ways 98
16 ways Q9
16 ways J9

Flush draws:

Q high: I'd say about 3 tops (QJ/Q9/Q8)
J high: about 2, maybe 3
And let's say 3 for 98, 86, and 65.

So I'm looking good on the flop: I'd guess there are 10 ways I'm ahead for every 1 way I'm behind.

(sometimes I just round to about 5 to make counting easier, though it's less accurate. So, 8 ways becomes 10, 12 ways becomes, 10, 6 ways becomes 5, etc. I'm not Rain Man after all)

On the turn, I'm looking worse, but still ahead:

Ways I'm behind increases to about 37

I think he dumps some of his flush draws when the board pairs. So I'll decrease those to 4.

River is ugly:

Now I'm behind everything but KJ/KT/98 and about 4 flush draws.

So there are about 80 ways he can beat me and I'd estimate about 25 ways I'm ahead (since I've showed strength and a K is in my range, I'd say he bluffs a low % of time). Of course I forgot about AJ. Throw that in there too.

Edited by derosnec (06/13/07 11:29 AM)


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Chomp
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Re: Combinatorics (very very long) [Re: derosnec]
      #10771454 - 06/13/07 11:34 AM

Custom title for Derosnec please: "Comboman".

Awesome stuff.


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Waingro
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Re: Combinatorics (very very long) [Re: derosnec]
      #10771609 - 06/13/07 11:47 AM

I really like your analysis in hand 2. In hand 1, it is really opponent specific. Against many you had the best hand if villain folded. Often they call with A high. Counting all ways a 60/40 can have a pair here vs the times he doesnīt seems cumbersome.

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orig!naL
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Re: Combinatorics (very very long) [Re: Bowlboy]
      #10771674 - 06/13/07 11:53 AM

Great post.

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bozzer
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Re: Combinatorics (very very long) [Re: Chomp]
      #10771809 - 06/13/07 12:04 PM

Quote:

Custom title for Derosnec please: "Comboman".

Awesome stuff.




what about 'Combinatrix'?


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bozzer
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Re: Combinatorics (very very long) [Re: bozzer]
      #11672564 - 08/14/07 08:17 PM

sticky please!!!

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