Tsao, Hung-ping (2021). Evolutionary Mathematics and Science for Two Aspects of Binomial Coefficients: Polynomial and Combination. In: “Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)”, Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 3, Number 8, August 2021; 42 pages. Lenox Institute Press, Auburndale, MA, 02466, USA. No. STEAM-VOL3-NUM8-AUG2021; ISBN 978-0-9890870-3-2. …………ABSTRACT: We make distinction of two important roles for binomial coefficients to play in Number Theory and Trigonometry, respectively. The polynomial aspect of C (n, k) enables the derivation of both explicit and implicit formulas for sums of powers of arithmetic progressions, whereas its combinatorial aspect helps the derivation of multiple angle formulas for Sine, Cosine and Tangent functions. ……….KEYWORDS: Polynomial expression, Sorting, Commutative ring, Stirling number, Eulerian number, Natural sequence, Powered sum, Binomial coefficient, Pascal triangle, Bernoulli coefficient, Arithmetically progressive sequence, Recursion, Iteration, Trigonometry, Multi angle formula Sine, Cosine, Tangent, De Moivre’s Theorem.