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Discrete Math by R.S. Chang, Dept. CSIE, NDHU1 Chapter 8 + or -?

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU2 Chapter The Principle of Inclusion and Exclusion 不滿足 c i 且不滿足 c j 不滿足 c i 或不滿足 c j

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU3 Chapter The Principle of Inclusion and Exclusion =N-[ + ] +

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU4 Chapter The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU5 Chapter The Principle of Inclusion and Exclusion Theorem 8.1 The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU6 Chapter The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU7 Chapter The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU8 Chapter The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU9 Chapter The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU10 Chapter 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.

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU11 Chapter The Principle of Inclusion and Exclusion

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU12 Chapter Generalizations of the Principle c1c1 c2c2 c3c E 1 : regions 2,3,4 E 2 : regions 5,6,7

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU13 Chapter Generalizations of the Principle

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU14 Chapter Generalizations of the Principle

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU15 Chapter Generalizations of the Principle

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU16 Chapter 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)

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU17 Chapter 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 d 7 therefore, 7! d 7

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU18 Chapter 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 r k or r k (C). idea: break up a large board into smaller subboards

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU19 Chapter Rook Polynomials C1C1 C2C2 C Did this occur by luck or is something happening here that we should examine more closely?

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU20 Chapter Rook Polynomials C1C1 C2C2 C To obtain r 3 for C: (a) All three rooks are on C 2 :(2)(1)=2 ways (b) Two on C 2 and one on C 1 :(10)(4)=40 (c) One on C 2 and two on C 1 :(7)(2)=14 total=(2)(1)+(10)(4)+(7)(2)=56

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU21 Chapter Rook Polynomials * decompose this board according to (*) CeCe CsCs put one at * * is empty * CsCs CeCe

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU22 Chapter Rook Polynomials * =x=x + ** =x2=x2 +x+x+x+x+ * =x2=x2 +2x+x+x +

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU23 Chapter 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) R 1 will not sit at T 1 or T 2 (b) R 2 will not sit at T 2 (c) R 3 will not sit at T 3 or T 4 (b) R 4 will not sit at T 4 or T 5 R1R1 R2R2 R3R3 R4R4 T1T1 T2T2 T3T3 T4T4 T5T5 It would be easier to work with the shaded area since it is less than the unshaded one. condition c i : R i is in a forbidden position

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU24 Chapter Arrangements with Forbidden Positions Ex R1R1 R2R2 R3R3 R4R4 T1T1 T2T2 T3T3 T4T4 T5T5 condition c i : R i is in a forbidden position

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU25 Chapter 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

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU26 Chapter Arrangements with Forbidden Positions c i : 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:

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Discrete Math by R.S. Chang, Dept. CSIE, NDHU27 Chapter 8 Exercise: P369: 16 P373: 8 P376: 14 P382: 5 End of chapter 8.

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