Tsao, Hung-ping (2022). Evolutionary mathematics and science for GENERAL BINOMIAL COEFFICIENTS: ALTERNATE SUMS OF ORTHOGONAL PRODUCTS OF STIRLING NUMBERS. In: “Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)”, Wang, Lawrence K. and Tsao, Hung-ping (editors). Volume 4, Number 4, April 2022; 31 pages. Lenox Institute Press, Auburndale, MA, 02466, USA. No. STEAM-VOL4-NUM4-APR2022; ISBN 978-0-9890870-3-2. ……………ABSTRACT: We shall investigate various alternate sums of orthogonal products of Stirling numbers. The Stirling numbers of the first and second kinds can be considered inverses of each other in light of two well-known matrix inverse relationships involving Kronecker delta. These two special alternate sums of orthogonal products of Stirling numbers trigger our further investigation. More importantly, the triangular arrays of combination (binomial coefficients) and permutation are among such alternate sums. From the prospective of general Stirling numbers, such seemingly wonderful coincidences are just matters of fact!…………KEYWORDS: Binomial coefficient, Stirling number, Alternate sum, Orthogonal product, Combination, Permutation, Natural sequence, Arithmetically progressive sequence.