L2 Supplementary Notes Page : Recap l General Product Principle n For counting list l Counting Functions n Functions are lists l Function concepts n Injection, surjection, and bijection l Bijiection Principle

L2 Supplementary Notes Page : Product Principle

L2 Supplementary Notes Page : Recap/Counting Functions

L2 Supplementary Notes Page : Injection and Surjection

L2 Supplementary Notes Page : Bijection

L2 Supplementary Notes Page : Bijection Principle

L2 Supplementary Notes Page : Today

L2 Supplementary Notes Page 8 Increasing Triples l Can we count using general product principle? n # of possible values for i?  n-2 n # of possible values for j?  Depends on value of i!  Violates condition of general product principle

L2 Supplementary Notes Page 9 Use Bijection Principle l n=4 l All increasing triples listed. l Do we have all the subsets? How about subset {2, 1, 3}? n It is the same as {1, 2, 3}

L2 Supplementary Notes Page 10 Increasing Triples

L2 Supplementary Notes Page 11 Increasing Triples

L2 Supplementary Notes Page 12 Increasing Triples l Same as n The number of 3-element subsets from {1, 2, …, n} n What is this number? l First consider another number n The number of 3-element permutations from {1, 2, …, n}

L2 Supplementary Notes Page 13 l Order matters, not required to be increasing l 3-element permutations from {1, 2, 3, 4}

L2 Supplementary Notes Page 14 l General product principle can be applied here although cannot for increasing triples. Why?

L2 Supplementary Notes Page 15 3-element subsets/3-element permutations

L2 Supplementary Notes Page 16 3-element subsets/Increasing triples l Same as n The number of 3-element subsets from {1, 2, …, n}

L2 Supplementary Notes Page 17

L2 Supplementary Notes Page 18 Permutations l Number of k-element permutations l Number of permutations of a set of size n

L2 Supplementary Notes Page 19 K-th falling factorial

L2 Supplementary Notes Page 20

L2 Supplementary Notes Page 21 k-element subsets/k-elemen permutations

L2 Supplementary Notes Page 22 k-element subsets/k-elemen permutations

L2 Supplementary Notes Page 23

L2 Supplementary Notes Page 24 Summary l Number of increasing triples: l Number of 3-element subsets l Number of 3-element permutations l Number of k-element subsets l Number of k-element permutations

L2 Supplementary Notes Page 25 Assignment 2: Problem 9 l Exco Members: Year 1: 4; Year 2: 5; Year 3: 3 l Total number of members: 12 l Choose 6 from 12

L2 Supplementary Notes Page 26 Assignment 2: Problem 9 l Exco Members: Year 1: 4; Year 2: 5; Year 3: 3 l Number of ways to choose Treasury Subcommittee n 12 choose 3 l Number of ways to choose Events Subcommittee n 9 choose 3

L2 Supplementary Notes Page 27 Assignment 2: Problem 9 l Exco Members: Year 1: 4; Year 2: 5; Year 3: 3 l Number of ways to choose Treasury Subcommittee l Number of ways to choose Events Subcommittee n 9 choose 3 l Answer

L2 Supplementary Notes Page 28 Assignment 2: Problem 9 l Exco Members: Year 1: 4; Year 2: 5; Year 3: 3 l “At least” makes things complex.

L2 Supplementary Notes

Wrong Answer l First choose 1 from each year l Then pick 3 from remaining 9 members l Answer

L2 Supplementary Notes Wrong Answer l First choose 1 from each year n Anthony (Y1), Mike (Y2), William (Y3) l Then pick 3 from remaining 9 members n Billy (Y1), Calvin(Y1), David (Y1), l Committee n Anthony (Y1), Billy (Y1), Calvin(Y1), David (Y1), Mike (Y2), William (Y3)

L2 Supplementary Notes Wrong Answer l First choose 1 from each year n Billy (Y1), Mike (Y2), William (Y3) l Then pick 3 from remaining 9 members n Anthony (Y1), Calvin(Y1), David (Y1), l Same Committee as before n Anthony (Y1), Billy (Y1), Calvin(Y1), David (Y1), Mike (Y2), William (Y3) l The same committee counted twice. n Not a bijection