How did Gregor Mendel use mathematics to analyze inheritance patterns?

Gregor Mendel revolutionized the study of genetics by applying his training in mathematics and physics to biological experimentation. Unlike his predecessors, Mendel grew statistically significant populations of garden peas and tracked individual characters separately to establish their statistical relations.1

By observing the proportions of traits in successive generations, he identified consistent ratios, such as the 3:1 ratio of dominant to recessive traits in the second generation (F${}_2$). He recognized that these results could be represented using combinatorial mathematics, specifically the expansion of the binomial equation:21
$$(A+a{)}^2=A^2+2Aa+a^2$$
In this model, A represents the dominant characteristic and a the recessive, resulting in a 1:2:1 ratio. When analyzing multiple traits simultaneously, he combined these binomial equations to demonstrate that traits are transmitted independently, a principle now known as the law of independent assortment. His rigorous methodology established a foundational prototype for modern genetic research.12