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8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion
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8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion
Theorem 8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion

8.1 The Principle of Inclusion and Exclusion
Ex. 8.6 Construct roads for 5 villages such that no village will be isolated. In how many ways can we do this? O.K. not O.K.

8.1 The Principle of Inclusion and Exclusion

8.2 Generalizations of the Principle
E1: regions 2,3,4 E2: regions 5,6,7 1 2 6 5 8 4 7 3 c3 c2

8.2 Generalizations of the Principle

8.2 Generalizations of the Principle

8.2 Generalizations of the Principle

8.3 Derangements: Nothing Is in Its Right Place
Ex. 8.8 Find the number of permutations such that 1 is not in the first place, 2 is not in the second place, ..., and 10 is not in the tenth place. (derangements of 1,2,3,...,10)

8.3 Derangements: Nothing Is in Its Right Place
Ex Assign 7 books to 7 reviewers two times such that everyone gets a different book the second times. Ans: first time 7!, second time d7 therefore, 7! d7

8.4 Rook Polynomials rook: castle Determine the number of ways in which k rooks can be placed on the chessboard so that no two of them can take each other, i.e., no two of them are in the same row or column of the chessboard. Denote this number by rk or rk(C). 3 2 1 4 5 6 idea: break up a large board into smaller subboards

8.4 Rook Polynomials C1 Did this occur by luck or is something happening here that we should examine more closely? C2 C

8.4 Rook Polynomials To obtain r3 for C: (a) All three rooks are on C2:(2)(1)=2 ways (b) Two on C2 and one on C1:(10)(4)=40 (c) One on C2 and two on C1:(7)(2)=14 total=(2)(1)+(10)(4)+(7)(2)=56 C1 C2 C

8.4 Rook Polynomials * decompose this board according to (*) * put one at * Ce * is empty Ce Cs Cs

8.4 Rook Polynomials * * * =x + * =x2 +x +x + =x2 +2x +x +

8.5 Arrangements with Forbidden Positions
Ex Arrange 4 persons to sit at five tables such that each one sits at a different table and with the following conditions satisfied: (a) R1 will not sit at T1 or T2 (b) R2 will not sit at T2 (c) R3 will not sit at T3 or T4 (b) R4 will not sit at T4 or T5 T1 T2 T3 T4 T5 It would be easier to work with the shaded area since it is less than the unshaded one. R1 R2 R3 R4 condition ci: Ri is in a forbidden position

8.5 Arrangements with Forbidden Positions
Ex. 8.11 T1 T2 T3 T4 T5 R1 R2 condition ci: Ri is in a forbidden position R3 R4

8.5 Arrangements with Forbidden Positions
Ex We have a pair of dice; one is red, the other green. We roll these dice six times. What is the probability that we obtain all six values on both the red dice and the green die if we know that the ordered pairs (1,2), (2,1),(2,5),(3,4),(4,1),(4,5), and (6,6) did not occur? [(x,y) indicates x on the red die and y on the green.] Relabeling the rows and columns 1 1 2 2 3 4 4 3 5 5 6 6

8.5 Arrangements with Forbidden Positions
ci: the condition where, having rolled the dice six times, we find that all six values occur on both the red die and the green die, but i on the red die is paired with one of the forbidden numbers on the green die Then the number of ordered sequences of the six rolls of the dice for the event we are interested in is:

Exercise: P369: 16 P373: 8 P376: 14 P382: 5 End of chapter 8.