Chapter 5: Permutations and Combinations

The material distinguishes between permutations, which represent ordered arrangements where sequence matters, and combinations, which represent unordered selections where sequence is irrelevant. The factorial function serves as the primary computational tool, defined as the product of an integer and all positive integers below it, with the conventional definition that zero factorial equals one. For permutations of distinct objects, the total count equals n factorial. When objects contain repetitions, such as repeated letters in a word, the calculation requires dividing by the factorials of each repeated element's frequency to avoid overcounting identical arrangements. Permutations with restrictions require strategic analysis of constrained positions before addressing unrestricted ones, with special techniques for keeping objects together or forcing them apart. When selecting r objects from n available objects for arrangement, the permutation formula becomes n factorial divided by the quantity n minus r factorial. Combinations eliminate the significance of order, so the formula incorporates division by r factorial to account for redundant orderings. Multiple selection problems follow additive and multiplicative principles depending on whether scenarios are mutually exclusive or independent. The chapter demonstrates how these counting techniques directly support probability calculations by providing the numerator of favorable outcomes and denominator of total possible outcomes in equiprobable event spaces. This systematic approach to enumeration substantially reduces computational complexity in solving complex probability problems without requiring exhaustive listing or extensive tree diagrams.