This paper presents the relations of optimized combination with traditional combination and permutation. Combinatorics is a collection of several computing techniques used in science and management. In this research paper, new results on both combination and permutation are introduced for the researchers who are involving to find solutions for the scientific problems.

This paper presents innovative series and summations derived from the optimized combinations relating to the combinatorics. These series and summations will be useful for the researchers who are involving to solve the scientific problems.