Last time I was trying to explain hypergeometric distribution (describing probability of k successes in n draws from a finite population without replacement). Maybe that was bit too steep to start this series of mathematical topics, so I will take one step back and talk about combinations, and the binomial coefficient.
In mathematics a combination is a way of selecting several things out of a larger group where (unlike permutations) order does not matter. Combinations can refer to the combination of n things taken k at a time with or without repetitions. In this article we are only looking at combinations without repetitions. For any set S containing n elements, the number of distinct k-element subsets of it that can be formed (the so-called k-combinations of its elements) is given by the binomial coefficient.
In smaller cases it is possible to count the number of combinations. For example, take a look at a set of the first for integer number (1,2,3,4) and subsets consisting of two (different) members of this set. You can make 6 subsets of 2 numbers: (1;2), (1;3), (1;4), (2;3), (2;4), (3;4). For larger sets this explicit enumeration can become very tedious.
The formula for the binomial coefficient (or short binom) is given (for 0 ≤ k ≤ n) as:
The binom is often read as "n choose k", and the factorial n! (of a non-negative integer n) is the product of all positive integers less than or equal to n. So you can extend the formula for the binom as:
Texas Hold 'Em stands. That's quite easy to calculate with the formula above. There are 52 different cards in Poker, and with a starting hand of 2 cards, you get 1326 different starting hands:
References: (all Wikipedia)