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MAT 2720 Discrete Mathematics Section 6.1 Basic Counting Principles http://myhome.spu.edu/lauw General Goals Develop counting techniques. Set up a framework for solving counting problems. The key is not (just) the correct answers. The key is to explain to your audiences how to get to the correct answers (communications). Goals Basics of Counting • Multiplication Principle • Addition Principle • Inclusion-Exclusion Principle Example 1 License Plate LLL-DDD # of possible plates = ? Analysis License Plate LLL-DDD # of possible plates = ? Procedure: Step 1: Step 2: Step 3: Step 4: Step 5: Step 6: Multiplication Principle Suppose a procedure can be constructed by a series of steps Step 1 Step 2 Step 3 Step k n1 ways n2 ways n3 ways nk ways Number of possible ways to complete the procedure is n1 n2 nk Example 2(a) Form a string of length 4 from the letters A, B, C , D, E without repetitions. How many possible strings? Example 2(b) Form a string of length 4 from the letters A, B, C , D, E without repetitions. How many possible strings begin with B? Example 3 Pick a person to joint a university committee. 37 Professors 83 Students EE Department # of possible ways = ? Analysis Pick a person to joint a university committee. 37 Professors 83 Students EE Department # of possible ways = ? The 2 sets: : Addition Principle X1 X2 X3 Xk n1 elements n2 elements n3 elements nk elements Number of possible element that can be selected from X1 or X2 or …or Xk is n1 n2 nk OR X1 X 2 X k n1 n2 nk Example 4 A 6-person committee composed of A, B, C , D, E, and F is to select a chairperson, secretary, and treasurer. chairperson secretary treasurer Committee A,B,C,D,E,F Example 4 (a) In how many ways can this be done? chairperson secretary treasurer Committee A,B,C,D,E,F Example 4 (b) In how many ways can this be done if either A or B must be chairperson? chairperson secretary treasurer Committee A,B,C,D,E,F Example 4 (c) In how many ways can this be done if E must hold one of the offices? chairperson secretary treasurer Committee A,B,C,D,E,F Example 4 (d) In how many ways can this be done if both A and D must hold office? chairperson secretary treasurer Committee A,B,C,D,E,F Recall: Intersection of Sets (1.1) The intersection of X and Y is defined as the set X Y x | x X and x Y X Y X Y Recall: Intersection of Sets (1.1) The intersection of X and Y is defined as the set X Y x | x X and x Y X 1, 2,3 , Y 3, 4,5 X Y X Y 3 4 1 2 X 3 Y 5 Example 5 What is the relationship between X , Y , X Y , and X Y ? X Y X Y 4 1 2 X 3 Y 5 X 1, 2,3 X Y 3, 4,5 Y X Y 3 X Y X Y 1, 2,3, 4,5 X Y Inclusion-Exclusion Principle X Y X Y X Y X Y X Y X Y Example 4(e) How many selections are there in which either A or D or both are officers?. chairperson secretary treasurer Committee A,B,C,D,E,F Remarks on Presentations Some explanations in words are required. In particular, when using the Multiplication Principle, use the “steps” to explain your calculations A conceptual diagram may be helpful. MAT 2720 Discrete Mathematics Section 6.2 Permutations and Combinations Part I http://myhome.spu.edu/lauw Goals Permutations and Combinations • Definitions • Formulas • Binomial Coefficients Example 1 6 persons are competing for 4 prizes. How many different outcomes are possible? 1st prize 3rd prize 2nd prize D Step 1: Step 2: Step 3: Step 4: F 4th prize C E B A r-permutations A r-permutation of n distinct objects x1, x2 , is an ordering of an r-element subset of x1, x2 , , xn 1st 2nd x1 x2 3rd x3 r - th xn , xn r-permutations A r-permutation of n distinct objects x1, x2 , is an ordering of an r-element subset of x1, x2 , , xn The number of all possible ordering: P(n, r ) 1st 2nd x1 x2 3rd x3 r - th xn , xn Example 1 6 persons are competing for 4 prizes. How many different outcomes are possible? 1st prize D P(6, 4) 3rd prize 2nd prize F 4th prize C E B A Theorem P(n, r ) n (n 1) (n 2) n! (n r )! 1st 2nd x1 x2 3rd x3 (n r 1) r - th xn Example 2 100 persons enter into a contest. How many possible ways to select the 1st, 2nd, and 3rd prize winner? Example 3(a) How many 3-permutations of the letters A, B, C , D, E, and F are possible? Example 3(b) How many permutations of the letters A, B, C , D, E, and F are possible. Note that, “permutations” means “6permutations”. Example 3(c) How many permutations of the letters A, B, C , D, E, and F contains the substring DEF? Example 3(d) How many permutations of the letters A, B, C , D, E, and F contains the letters D, E, and F together in any order?