The Foundation To Mathematical Sciences comprises of the corrections/solutions giving by the Author in the Series of Negative and Fractional Binomial which comprises of a Mistake Made by Most scientist for so many years . In the Chapter of the Negative Binomial the Author generated the series of the Negative Triangle which he named after him called the Anekwe's Triangle in the series of Negative Binomial just like that of positive binomial which has a triangle called the pascals Triangle. In the second chapter the Author generated three set of General Formular named after him for solving special forms of integration which he applied to solve problems in terms of Statistical and Thermal Physics. In the last chapter on Equations in permutation and combinations the Author generated a method for solving problems in equations in permutation and combinations involving r number of unknowns, which he called Anekwe's Method of swapping factorials and many more scientific discoveries whichwas explicitly discuss in this book. This book is the Basics/The Foundation upon which mathematical Sciences is built.