97 | P a g e DISCRETE MATHEMATICS Activity 5a 1. A student can choose a computer project from one of three lists. The three lists contain 23, 15 and 19 possible projects respectively. No project is on more than one list. How many possible projects are there to choose from? 2. How many different license plates are available if each plate contains a sequence of three letters followed by three digits (and no sequences of letters are prohibited, even if they are obscene)? 3. Suppose a bookcase shelf has 5 History texts, 3 Sociology texts, 6 Anthropology texts and 4 Psychology texts. Find the number of of ways a student can choose: a. One of the texts b. One of each type of texts 4. A mathematics class contains 8 male students and 6 female students. Find the number of ways that the class can elect: a. 1 class representative b. 2 class representative with 1 male and 1 female c. 1 president and 1 vice president 5. Suppose a code consists of five characters, 2 letters followed by 3 digits. Find the number of: a. Codes b. Codes with distinct letter c. Codes with the same letters d. Codes with distinct digit
98 | P a g e DISCRETE MATHEMATICS 5.2 Compute permutations and combinations 5.2.1 Describe permutations with and without repetition • A permutation of a set of distinct objects taken at a time without repetition is an arrangement of objects in a specific order. • The arrangement of these elements count. • The number of -permutations of a set with distinct elements is denoted by(, ). • We can calculate (, ) with the product rule: (, ) = ( − 1)( − 2). . . . ( − + 1) ( choices for the first element, ( − 1) for the second one, ( − 2) for the third one…). • General Formula : ❖ Permutation Without Repetition ❖ Permutation With Repetition n, r ! P ( )! n n r = − r n
99 | P a g e DISCRETE MATHEMATICS 5.2.2 Solve problems by using permutations Example 1: In how many ways can you arrange the 3 objects in the set {A, B, C} without repetition? Solution: To arrange all 3 objects in the set {A, B, C}, 6 3! 3 2 1 = = List all of them: ABC, ACB, BAC, BCA, CAB, CBA Example 2: How many ways are there to select a first prize winner, a second winner, and a third prize winner from 100 different people who have entered a contest? Solution: Because it does matter which person wins which prizes, we use permutations. The number of ways to pick the 3 prize winner from 100 people: ( ) 970200 100,3 100 99 98 = P = Example 3: How many permutations of the letter ABCDEFGH contain the string ABC (ABC occur as a block)? Solution: Because the letter ABC must occur as a block, we can find the number of permutations of six objects: Block ABC and individual letter (D, E, F, G, H) Hence, there are 720 permutations of the letter ABCDEFGH in which ABC occurs as a block. ABC D E F G H 6! = 720
100 | P a g e DISCRETE MATHEMATICS Activity 5b 1. Find the value of each of these quantities by using formula: a. (6,4) b. (9,5) c. (8,3) 2. How many possibilities are there for the first, second and third place positions in a car race with 12 cars if all orders of finish are possible? 3. How many permutations of the letters ABCDEFG contain: a. The string BCD b. The string CFGA c. The strings BA and GF d. The strings ABC and DE e. The strings ABC and CDE 4. Find the number of distinct permutations that can be formed from all the letters of each word: a. THOSE b. UNUSUAL c. SOCIOLOGICAL d. PROPOSITION 5. A class contains 8 students. Find the number of samples of size 3: a. With replacement b. Without replacement 6. A debating team consists of 3 boys and 3 girls. Find the number of ways they can sit in a row where: a. There are no restriction b. The boys and girls are each to sit together c. Just the girls are to sit together
101 | P a g e DISCRETE MATHEMATICS 5.2.3 Describe Combinations with and without repetition • A combination of a set of distinct objects taken at a time without repetition is an -element subset of the objects. • The arrangement of elements does not count. • The number of combinations of distinct objects taken at a time without repetition is given by: • General Formula: ❖ Combination Without Repetition ❖ Combination With Repetition n, r ! C !( )! n n r r n r = = − ( ) ( ) ( ) !( 1)! 1 ! 1, 1, 1 − + − = + − = + − − r n n r C n r r C n r n
102 | P a g e DISCRETE MATHEMATICS 5.2.4 Solve problems by using combinations Example 1: How many committees of three can be formed from seven people? Solution: Each committee is a combination of seven people taken three at a time. Thus, the number of committees can be formed is: ( ) ( ) 35 3 2 1 4 3 2 1 7 6 5 4 3 2 1 3! 7 3 ! 7! ! ! ! 7 3 = = − = − = C r n r n C n r Example 2: In how many ways can we select a committee of two women and three men from a group of five distinct women and six distinct men? Solution: ⚫ The committee can be constructed in two ways; i. Select the women ii. Select the men ⚫ To select the women; Combination of two women can be selected from five women is
103 | P a g e DISCRETE MATHEMATICS ( ) 10 12 120 2 1 3 2 1 5 4 3 2 1 2! 5 2 ! 5! 5 2 = = = − C = ⚫ To select the men; Combination of three men can be selected from six men is ( ) 20 36 720 3 2 1 3 2 1 6 5 4 3 2 1 3! 6 3 ! 6! 6 3 = = = − C = Hence, by the multiplication Rule, the total number of committee that can be formed is: 10 × 20 = 200 Example 3: Suppose that a cookie shop has four different kinds of cookies. How many different ways can six cookies be chosen? Assume that only the type of cookie, and not the individual cookies or the order in which they are chosen, matters. Solution: ⚫ The number of ways to choose six cookies is the number of 6 – combinations of a set with four elements.
104 | P a g e DISCRETE MATHEMATICS ( ) ( ) ( ) ( ) 84 1 2 3 9 8 7 9,6 9,3 4 6 1, 6 4 6 1, 4 1 = = = + − = + − − C C C C Combinations & Permutations With & Without Repetition Type Repetition Allowed? Formula - permutations No - combinations No (, ) = ! ! ( − )! - permutations Yes - combinations Yes ( ) ( ) ( ) !( 1)! 1 ! 1, 1, 1 − + − = + − = + − − r n n r C n r r C n r n n, r ! P ( )! n n r = −
105 | P a g e DISCRETE MATHEMATICS Activity 5c 1. Find the value of each of these quantities by using formula: a. (5,1) b. (8,4) c. (7,3) d. (12,6) 2. In how many ways can a set of five letters can be selected from the alphabet? 3. A box contains 8 blue socks and 6 red socks. Find the number of ways two socks can be drawn from the box if: a. They can be any colour b. They must be the same colour 4. How many bit strings of length 10 contain: a. Exactly four 1s b. At most four 1s c. At least four 1s d. An equal number of 0s and 1s 5. A class contains 9 men and 3 women. Find the number of ways a teacher can select a committee of 4 from the class where there is: a. No restrictions b. 2 men and 2 women c. Exactly one woman d. At least one woman 6. A coin is flipped 10 times where each flip comes up either heads or tails. How many possible outcomes: a. Are there in total b. Contain exactly two heads c. Contain at most three tails d. Contain the same number of heads and tails Scan this QR code to get the previous Final Examination Question related to this chapter
106 | P a g e DISCRETE MATHEMATICS REFERENCES Kevin, F. (2017). Discrete Mathematics and Applications(2nd). Chapman and Hall (GB). Richard, J. (2017). Discrete Mathematics, (8 th Edition). USA: Pearson. Rosen, K. H. (2018). Discrete Mathematics and Its Applications, 8th Edition. USA: McGraw-Hill Education. Seymour Lipschutz and Marc Lipson. (2007). Theory and Problems of Discrete Mathematics, 3 rd Edition. USA: McGraw-Hill Education. Susanna S. Epp. (2010). Discrete Mathematics with Applications, 4th Edition. Canada: Richard Stratton. Yeow Pow Choo, Thavamani A/P Renu, Kamalah A/P Raman, Wong Jin Wen and Vincent De Selva A/L Santhanasamy. (2019). Buku Teks KSSM Matematik Tingkatan 4. Malaysia: Pustaka Yakin Pelajar Sdn. Bhd.
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