l Combinatorics 61. Pigeonhole principle &o Pigeon and pigeonholes ☆ example, exercise
❖II Combinatorics ❖1. Pigeonhole principle ❖ Pigeon and pigeonholes ❖ example，exercise
☆2。 Permutations and combinations Permutations of sets combinations of sets 8 circular permutation Permutations and combinations of multisets ◆ Formulae &inclusion-exclusion principle generating functions &integral solutions of the equation
❖2. Permutations and Combinations ❖ Permutations of sets, Combinations of sets ❖ circular permutation ❖ Permutations and Combinations of multisets ❖ Formulae ❖ inclusion-exclusion principle ❖ generating functions ❖ integral solutions of the equation
g Applications of Inclusion-Exclusion principle ☆ example, exercise Applications generating functions and Exponential generating functions 冷ex=1+x+x2/2!++x/n!+. 令x+x2/2!+,+x/n!+,=ex-1 冷e=1-x+x2/2!+…+(-1)x"/n!+…; 冷1+x212!+,+x2n(2n)!+,=(e+ex)/2; 冷x+x3!+…+x2n+1(2n+1)!+…=(ex-e-x)/2; %o examples, and exercises 3. recurrence relation 4 Using Characteristic roots to solve recurrence relations Using Generating functions to solve recurrence relations .examples, and exercises
❖ Applications of Inclusion-Exclusion principle ❖ example,exercise ❖ Applications generating functions and Exponential generating functions ❖ e x=1+x+x2 /2!+…+xn /n!+…; ❖ x+x2 /2!+…+xn /n!+…=ex -1; ❖ e -x=1-x+x2 /2!+…+(-1)nx n /n!+…; ❖ 1+x2 /2!+…+x2n/(2n)!+…=(ex+e-x )/2; ❖ x+x3 /3!+…+x2n+1/(2n+1)!+…=(ex -e -x )/2; ❖ examples, and exercises ❖ 3. recurrence relation ❖ Using Characteristic roots to solve recurrence relations ❖ Using Generating functions to solve recurrence relations ❖ examples, and exercises
Chapter 5 Graphs . the puzzle of the seven bridge in the Konigsberg, o on the Pregel B c 7777x D
Chapter 5 Graphs ❖ the puzzle of the seven bridge in the Königsberg, ❖ on the Pregel
B 画只日 D (b)的