Complete multinomial expansions Naiyang Ma Department of Metallurgical Engineering, University of Utah, 135 S 1460 E, Rm 412, Salt Lake City, UT 84108, USA Abstract A complete form of multinomial expansions is developed in this paper, which shows the detailed structure of each decomposed term. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Multinomial expansion; Complete multinomial expansion; Element; Group 1. Introduction Multinomial expansion has wide applications in theory of number, proba- bility theory and stochastic process. For general purposes, let us consider the following formula: C  X m i1 f i  ! n ; 1 where m and n are positive integers, and f i is an arbitrary expression. According to multinomial expansion theorem [1], formula 1) can be ex- panded into C  X n n 1 n 2  n m   f n 1 1 f n 2 2  f nm m ; 2 where the sum is over all m-tuples of nonnegative integers, n 1 ; n 2 ; ... ; n m that sum to n. Applied Mathematics and Computation 124 2001) 365±370 www.elsevier.com/locate/amc E-mail address: nma@mines.utah.edu N. Ma). 0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII:S0096-300300)00102-8