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Article 1 - Ways To Make A Hand 
The following tables give the odds of being dealt specific poker hands and the odds of drawing winning hands. Also given are the number of possible ways a particular poker hand can be made with a 52-card deck.
WAYS TO MAKE A HAND
The number of ways to make various poker hands is the basis for most statistical calculations in poker or video poker. Before the advent of computers, professional gamblers reckoned these numbers manually, or hired the services of a mathematician. If you don't know how it is done, the following examples may be enlightening:
Assuming a 52-card deck with no wild cards, let's see how many ways there are to make a straight flush. It is not too hard to visualize that, for a single suit, there are 10 ways. These are A-2-3-4-5, 2-3-4-5-6, 3-4- 5-6-7,4-5-6-7-8,5-6-7-8-9,6-7-8-9-10,7-8-9-10-J,8-9-10-J-Q, 9-10-J-Q-K, and 10-J-Q-K-A.
Multiply these 10 ways by 4 suits, and we arrive at a total of 40 ways. Four of these are royal flushes, so the number is usually given as 36 ways.
Another seemingly simple example would be four-of-a-kind (quad). Since there are four cards in every rank (one of each suit), and 13 ranks, there must be 13 unique quads. That is true, except we have to take into account the fifth card. If four cards out of the deck are used for the quad, there are 48 cards left, anyone of which could be the fifth card. Therefore, to finish the calculation, we have to multiply the 13 quads by the 48 possibilities for the fifth card. Thus, 13 times 48 equals 624 ways.
This is pretty straightforward arithmetic, but when you get to the lower hands the calculations get more complicated and it is easier to overlook something. If you try to calculate the number of straights, for instance, don't forget to subtract the 40 straight flushes.
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(Submitted on: Thu Jan 30, 2003)
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Article 2 - Odds of Making a Hand
Why is the number of ways to make a hand important? For one, it is the basis for calculating the odds of being dealt a particular hand - and this is very important information for the poker player. Before we can do any further calculations,we need to know how many different 5-card hands can be deald from a 52-card deck.This is calculated as follows:
52x51x50x49x48
------------------------- = 2,598,960 ways
5x4x3x2x1
A mathematician would express this with a computational tool called factorials, but we will keep it as simple as possible. Now we can proceed with odds calculations.
To determine the odds of getting two pairs on the initial deal, for example, just divide 2,598,960 by 123,552(from the first table). The result is approximately 21, which means there is one chance in 21 of getting two pairs. This is the method used to figure the odds for all the hands in the first table of this chapter.
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(Submitted on: Wed Jan 29, 2003)
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