A standard 52-card deck can produce 2,598,960 unique five-card poker hands.

The calculation of these hands relies on the mathematical principle of combinations, specifically '52 choose 5'. This is expressed by the formula n! / (r!(n-r)!), where n is the total items and r is the number being chosen. In this case, the calculation is 52! / (5! * 47!), which results in exactly 2,598,960 possibilities.This branch of mathematics is known as combinatorics and was significantly advanced by Blaise Pascal and Pierre de Fermat in the 17th century. Their work laid the foundation for modern probability theory, which is essential for understanding card game odds. While the total number of hands is large, the number of 'Royal Flush' combinations is only 4, making the odds of hitting one approximately 1 in 649,740.The diversity of hands explains why poker remains a game of skill and statistical analysis. Even with over 2.5 million combinations, players can categorize these into nine distinct hand rankings. Professionals use these fixed mathematical probabilities to determine their 'pot equity' and long-term strategy. Understanding these figures is the basis for all modern game theory in competitive card playing.