1. The Product Rule: Use & Misuse

2. Copyright © Peter Cappello 20112 The Product Rule If a procedure has 1.A 1 st stage with s 1 outcomes 2.A 2 nd stage with s 2 outcomes and the composite outcomes are distinct then The procedure has s 1 x s 2 composite outcomes.

3. Copyright © Peter Cappello 20113 Visualizing the Procedure –The procedure for constructing a composite outcome requires a selection at each stage. –Visualize all invocations of the procedure as a tree. Each level in the tree corresponds to a stage. Each leaf in the tree corresponds to a composite outcome: Each leaf must be distinct.

4. Copyright © Peter Cappello 20114 Example 1 Let B = { 0, 1 } and V = { a, e, i, o, u }. How many 1-to-1 functions, f, are there from B to V? Solution: 1.Select the vowel for f( 0 ) (5 choices). 2.Select the vowel for f( 1 ) (4 choices). Thus, there are 5x4 1-to-1 functions from B to V.

5. Copyright © Peter Cappello 20115 Example 1 continued Visualize the selection process as a tree. ae iou (a,e)(u,o)(e,a)(i,a)(o,a)(u,a) 1. Pick f(0) 2. Pick f(1)

6. Copyright © Peter Cappello 20116 Misuse of the Product Rule The set of 5 vowels has how many subsets of 2 letters? Erroneous solution: 1.Pick the 1 st letter (5 choices). 2.Pick the 2 nd letter (4 choices). There are 5 x 4 subsets of 2 letters. Not! –Visualize the selection process above as a tree. –The composite outcomes are not distinct. Each leaf appears twice (e.g., ae, ea) The same set of 2 vowels is counted twice. To use the product rule properly, it is necessary that: Each component of the composite outcome is associated with 1 stage of the selection process. If it cannot be so associated, the product rule is used incorrectly.

7. Copyright © Peter Cappello 20117 Subset Example continued Visualize the selection process as a tree. ae iou {a,e}{u,o}{e,a}{i,a}{o,a}{u,a} 1. Pick “1 st ” vowel 2. Pick “2 nd ”

8. Copyright © Peter Cappello 20118 Proper Use of the Sum Rule The subsets are pairwise disjoint. The union of the subsets includes every element that you want to count. Sound familiar?