How did Blaise Pascal and Pierre de Fermat contribute to combinatorics?

In the context of mathematics, the 17th-century French mathematicians Blaise Pascal and Pierre de Fermat are widely considered the founders of modern combinatorics, a field that emerged largely through their collaborative work on the theory of probability.1

Their contributions were primarily driven by their efforts to solve problems related to games of chance, such as the "problem of points," which concerns the fair division of stakes in an interrupted game. To address these challenges, they developed methods for calculating combinations and probabilities:23

• Pascal’s Triangle: Pascal’s work on the "arithmetic triangle"—a triangular array of binomial coefficients—provided a systematic way to calculate the number of combinations and group them to solve elementary problems. While the array was known to earlier mathematicians, Pascal’s formalization linked it to the binomial theorem and combinatorial analysis.24
• Methodological Approaches: Both mathematicians defined sets of symmetrical cases to determine outcomes. Fermat approached these problems by calculating the ratio of favorable outcomes to total possible outcomes, while Pascal introduced the concept of "expectation" to evaluate the value of a player's position.2

These foundational discoveries established the basis for enumerative combinatorics and influenced the development of probability theory.21