Most results on pattern containment deal more directly with pattern avoidance, or the enumeration and characterization of strings which avoid a given set of patterns. Little research has been conducted regarding the word size required for a word to contain all patterns of a given set of patterns. The set of patterns for which containment is sought in this thesis is the set of preferential arrangements of a given length. The term preferential arrangement denotes strings of characters in which repeated characters are allowed, but not necessary. Cardinalities for sets of all preferential arrangements of given lengths and alphabet sizes are found, as well as cardinalities for sets where reversals fall into the same equivalence class and for sets in higher dimensions. The minimum word length and the word length necessary for a strict superpattern to contain all preferential arrangements for alphabet sizes two and three are also detailed.