The determination of bounds for the number of maximal subgroups of a given index in a finite group is relevant to estimate the number of random elements needed to generate a group with a given property. This problem has been studied extensively in the context of group theory and its applications in computer science and cryptography. The goal of this research activity is to develop new methods and techniques to determine these bounds, with a focus on the interplay between group theory and combinatorics. By exploring the connections between these two areas, we aim to provide new insights and tools for understanding the structure of finite groups and their maximal subgroups.