\input zb-basic \input zb-matheduc \iteman{ZMATH 2010e.00502} \itemau{Leung, Issic K.C.; Ching, Wai-Ki} \itemti{Cardinality of binary operations: a remark on the ubiquitous sum.} \itemso{Far East J. Math. Educ. 3, No. 2, 127-143 (2009).} \itemab Summary: We establish the sufficient conditions to determine how many binary operations can possibly take place between any two arbitrary elements from a given set, provided that the operation is well defined. If we mark and collect each of such operations in another set $S$, we call the number $P_N$ the cardinality of the set $S$ of binary operations between any two elements for a given set of $N$ elements. We find that such number $P_N$ is closely related to the sum of consecutive numbers, the Ubiquitous Sum [{\it S. J. Bezuszka} and {\it M. Kenney}, That ubiquitous sum: Math. Teacher 98, No. 5, 316--321 (2005; ME 2007a.00386)]. In particular, $P_N$ is simply the combination of selecting from $N$ distinct objects, two at a time. This idea can be generated to look for the cardinality of a set of ternary operations. We have verified that this cardinality is the same as the combination of selecting from $N$ distinct objects, three at a time. The results can be generalized to derive the formulae of factorization, when $T_n = 1^n + 2^n + 3^n + \cdots + N^n, n = 1,2,3,\dots$. We also discuss how the formulae are applicable in mathematics pedagogy. \itemrv{~} \itemcc{H20 H40} \itemut{cardinality; binary operations; ubiquitous sum; combination} \itemli{http://pphmj.com/abstract/4254.htm} \end