How did combinatorial mathematics help Gregor Mendel predict pea-plant offspring?

Gregor Mendel utilized his background in mathematics, specifically combinatorial mathematics, to analyze the transmission of hereditary traits in pea plants.1

By representing dominant characteristics as A and recessive ones as a, Mendel recognized that the observed 1:2:1 ratio of offspring in the second generation (F${}_2$) corresponded to the expansion of the binomial equation: $(A+a{)}^2=A^2+2Aa+a^2$. This mathematical framework allowed him to predict the proportions of true-breeding and hybrid offspring accurately.1

Furthermore, Mendel applied these principles to test whether multiple traits were inherited independently. By combining binomial equations, he predicted the ratios for crosses involving two or three traits simultaneously, such as the 9:3:3:1 ratio observed in dihybrid crosses. This confirmed his hypothesis regarding the law of independent assortment, demonstrating that separate gene pairs are transmitted independently during gamete formation. His rigorous use of statistics and probability established the foundation for modern genetics.213