Lagrange expansion formula and its application in combinatorial mathematics

There is a uniqueness theorem for an implicit function. When a function satisfies certain conditions, the equation can determine a unique implicit function. However, we cannot determine the display expression of the implicit function according to this theorem. The Lagrange expansion formula just compensates for this disadvantage, which explicitly gives the display expression of the implicit function. This paper will give a simpler proof of the Lagrange expansion formula and provide examples to illustrate its application in combinatorial mathematics.