1. + or -?

2. 8.1 The Principle of Inclusion and Exclusion
不滿足ci且不滿足cj
不滿足ci或不滿足cj

3. 8.1 The Principle of Inclusion and Exclusion+ ] +

4. 8.1 The Principle of Inclusion and Exclusion

5. 8.1 The Principle of Inclusion and Exclusion
Theorem 8.1 The Principle of Inclusion and Exclusion

6. 8.1 The Principle of Inclusion and Exclusion

7. 8.1 The Principle of Inclusion and Exclusion

8. 8.1 The Principle of Inclusion and Exclusion

9. 8.1 The Principle of Inclusion and Exclusion

10. 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.

11. 8.1 The Principle of Inclusion and Exclusion

12. 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

13. 8.2 Generalizations of the Principle

14. 8.2 Generalizations of the Principle

15. 8.2 Generalizations of the Principle

16. 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)

17. 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

18. 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

19. 8.4 Rook Polynomials C1
Did this occur by luck or is something
happening here that we should examine
more closely? C2 C

20. 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

21. 8.4 Rook Polynomials *
decompose this board according to (*) *
put one at * Ce
* is empty Ce Cs Cs

22. 8.4 Rook Polynomials * * * =x + *
=x2 +x +x +
=x2
+2x +x +

23. 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

24. 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

25. 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

26. 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:

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