Permutations, Combinations and Sample Spaces

Lesson 17 of STA301 explores combinatorial analysis, detailing Permutations and Combinations as essential mathematical tools for enumerating sample spaces and computing event probabilities. Combinatorial calculations rely on Factorial Notation: n! represents the product of all positive integers from n down to 1, with the foundational mathematical convention 0! = 1. A Permutation is defined as an ordered arrangement of a subset of distinct objects. When selecting r objects from a collection of n distinct objects where order of arrangement matters, the number of permutations is given by nPr = n! / (n - r)!. The lecture also examines permutations of n objects where items are not all distinct: if there are n items with n1 of one kind, n2 of a second kind, up to nk of a kth kind, total distinct arrangements equal n! / [n1! * n2! * ... * nk!]. Furthermore, Circular Permutations—arranging n objects around a circle where relative positional rotation does not change the pattern—are evaluated as (n - 1)!. In contrast, a Combination represents an unordered selection of objects where sequence is irrelevant. The number of combinations of n distinct objects taken r at a time is given by nCr = n! / [r! * (n - r)!]. Permutations and combinations are linked by nPr = r! * nCr. Combinations satisfy fundamental algebraic identities: nCr = nC(n-r) and Pascal's Rule nCr + nC(r-1) = (n+1)Cr. The lecture applies these counting rules to categorize events within sample spaces: Mutually Exclusive (Disjoint) Events cannot occur simultaneously (A cap B = phi); Exhaustive Events collectively span the entire sample space (A1 U A2 U ... U Ak = S); and Equally Likely Events possess identical chances of occurrence due to physical symmetry, such as fair dice or balanced coins. In Virtual University midterm examinations, permutations of word letters, combination card selections, and mutually exclusive event identifications appear frequently. Reviewing sta301 handouts and practicing solved MCQs by Moaaz reinforces these concepts.