Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 85 4.13. Determine the validity of the following argument: If 7 is less than 4, then 7 ...

86 LOGIC AND PROPOSITIONAL CALCULUS [CHAP. 4 4.18. Letp(x)denote the sentence “x+ 2 >5.” State whether or notp(x)is a proposi ...

CHAP. 4] LOGIC AND PROPOSITIONAL CALCULUS 87 4.26. Negate each of the following statements: (a) If the teacher is absent, then s ...

CHAPTER 5 Techniques of Counting 5.1Introduction This chapter develops some techniques for determining, without direct enumerati ...

CHAP. 5] TECHNIQUES OF COUNTING 89 EXAMPLE 5.1 Suppose a college has 3 different history courses, 4 different literature courses ...

90 TECHNIQUES OF COUNTING [CHAP. 5 Binomial Coefficients The symbol ( n r ) , read “nCr”or“nChooser,” whererandnare positive int ...

CHAP. 5] TECHNIQUES OF COUNTING 91 Fig. 5-1 Pascal’s triangle Theorem 5.3: ( n+ 1 r ) = ( n r− 1 ) + ( n r ) 5.4Permutations Any ...

92 TECHNIQUES OF COUNTING [CHAP. 5 Corollary 5.5: There aren! permutations ofnobjects (taken all at a time). For example, there ...

CHAP. 5] TECHNIQUES OF COUNTING 93 (2) Sampling without replacement Here the element is not replaced in the setSbefore the next ...

94 TECHNIQUES OF COUNTING [CHAP. 5 Theorem 5.7: C(n, r)= P (n, r) r! = n! r!(n−r)! Recall that the binomial coefficient ( n r ) ...

CHAP. 5] TECHNIQUES OF COUNTING 95 5.7The Inclusion–Exclusion Principle LetAandBbe any finite sets. Recall Theorem 1.9 which tel ...

96 TECHNIQUES OF COUNTING [CHAP. 5 Fig. 5-2 (b) Mark and Erik are to play a tennis tournament. The first person to win two games ...

CHAP. 5] TECHNIQUES OF COUNTING 97 (b) 7! 10! = 7! 10 · 9 · 8 · 7! = 1 10 · 9 · 8 = 1 720 . 5.3. Simplify: (a) n! (n− 1 )! ;(b) ...

98 TECHNIQUES OF COUNTING [CHAP. 5 5.8. A history class contains 8 male students and 6 female students. Find the numbernof ways ...

CHAP. 5] TECHNIQUES OF COUNTING 99 (c) n= 12! 3! 2! 2! 2! , since there are 12 letters of which 3 areO, 2 areC, 2 areI, and 2 ar ...

100 TECHNIQUES OF COUNTING [CHAP. 5 PIGEONHOLE PRINCIPLE 5.19. Find the minimum numbernof integers to be selected fromS={ 1 , 2 ...

CHAP. 5] TECHNIQUES OF COUNTING 101 TREE DIAGRAMS 5.25. TeamsAandBplay in a tournament. The first team to win three games wins t ...

102 TECHNIQUES OF COUNTING [CHAP. 5 Now the term in the product which containsbris obtained from b[(nr− 1 )an−r+^1 br−^1 ]+a[(nr ...

CHAP. 5] TECHNIQUES OF COUNTING 103 SupplementaryProblems FACTORIAL NOTATION, BINOMIAL COEFFICIENTS 5.31. Find: (a) 10!, 11!, 12 ...

104 TECHNIQUES OF COUNTING [CHAP. 5 PERMUTATIONS WITH REPETITIONS, ORDERED SAMPLES 5.51. Find the number of permutations that ca ...