Tsao, Hung-ping 曹 恆 平 (2021). Evolutionary mathematics and science for General Binomial Coefficients and Fibonacci Numbers. In: “Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)”, Wang, Lawrence K. 王 抗 曝 and Tsao, Hung-ping 曹 恆 平 (editors). Volume 3, Number 4, April 2021; 34 pages. Lenox Institute Press, Newtonville, NY, 12128-0405, USA. No. STEAM-VOL3-NUM4-APR2021; ISBN 978-0-9890870-3-2. ————ABSTRACT: We first generalize binomial coefficients from the natural sequence based to arithmetically progressive sequences based and then display the key roles they play in the derivation of polynomial expressions of powered sums. We further come up with the new Fibonacci numbers based on arithmetically progressive sequences. As a matter of fact, the usual Fibonacci numbers are the upward diagonal sums of the Pascal triangular array, whereas these newly defined Fibonacci numbers are the upward diagonal sums of any triangular array.————KEYWORDS: Binomial coefficient, Pascal triangle, Fibonacci number, Natural sequence, Arithmetically progressive sequence, Stirling number, Eulerian number, Recursive formula, Upward diagonal, Fibonacci value.