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- 1. Set, Combinatorics, Probability & Number Theory Mathematical Structures for Computer Science Chapter 3 Copyright © 2006 W.H. Freeman & Co. w/modiﬁcations by D. Hyland-Wood, UMW Set, Combinatorics, Probability & Number Theory Thursday, March 11, 2010
- 2. Permutations ● An ordered arrangement of objects is called a permutation. ■ Hence, a permutation of n distinct elements is an ordering of these n elements. ● It is denoted by P(n,r) or nPr. ● Ordering of last four digits of a telephone number if digits are allowed to repeat ■ 10*10*10*10 = 10000 ■ Ordering of four digits if repetition is not allowed = 10*9*8*7 = 5040 = 10!/6! where n! = n*(n-1)*(n-2)*….*3*2*1 and by deﬁnition 0! = 1 ● Hence, mathematically, for r ≤ n, an r-permutation from n objects is deﬁned by ● P(n,r) = n*(n-1)*(n-2)*…..*(n-r+1) ⇒ P(n,r) = for 0 ≤ r ≤ n ● Hence, P(10,4) = 10! / (10-4)! = 10!/6! = 5040 Section 3.4 Permutations and Combinations 2 Thursday, March 11, 2010
- 3. Permutations: Some special cases ● P(n,0) = n! / n! = 1 ● This means that there is only one ordered arrangement of 0 objects, called the empty set. ● P(n,1) = n!/ (n-1)! = n ● There are n ordered arrangements of one object (i.e. n ways of selecting one object from n objects). ● P(n,n) = n!/(n-n)! = n!/0! = n! ● This means that one can arrange n distinct objects in n! ways, that is nothing but the multiplication principle. Section 3.4 Permutations and Combinations 3 Thursday, March 11, 2010
- 4. Permutation Examples 1. Ten athletes compete in an Olympic event. Gold, silver and bronze medals are awarded to the ﬁrst three in the event, respectively. How many ways can the awards be presented? 2. How many ways can six people be seated on six chairs? 3. How many permutations of the letters ABCDEF contain the letters DEF together in any order? 4. The professor’s dilemma: how to arrange four books on OS, seven on programming, and three on data structures on a shelf such that books on the same subject must be together? Section 3.4 Permutations and Combinations 4 Thursday, March 11, 2010
- 5. Permutation Examples 1. Ten athletes compete in an Olympic event. Gold, silver and bronze medals are awarded to the ﬁrst three in the event, respectively. How many ways can the awards be presented? Hence, 3 objects from a pool of 10 = P(10,3) = 720 2. How many ways can six people be seated on six chairs? 3. How many permutations of the letters ABCDEF contain the letters DEF together in any order? 4. The professor’s dilemma: how to arrange four books on OS, seven on programming, and three on data structures on a shelf such that books on the same subject must be together? Section 3.4 Permutations and Combinations 4 Thursday, March 11, 2010
- 6. Permutation Examples 1. Ten athletes compete in an Olympic event. Gold, silver and bronze medals are awarded to the ﬁrst three in the event, respectively. How many ways can the awards be presented? Hence, 3 objects from a pool of 10 = P(10,3) = 720 2. How many ways can six people be seated on six chairs? P(6,6) = 6! = 720 3. How many permutations of the letters ABCDEF contain the letters DEF together in any order? 4. The professor’s dilemma: how to arrange four books on OS, seven on programming, and three on data structures on a shelf such that books on the same subject must be together? Section 3.4 Permutations and Combinations 4 Thursday, March 11, 2010
- 7. Permutation Examples 1. Ten athletes compete in an Olympic event. Gold, silver and bronze medals are awarded to the ﬁrst three in the event, respectively. How many ways can the awards be presented? Hence, 3 objects from a pool of 10 = P(10,3) = 720 2. How many ways can six people be seated on six chairs? P(6,6) = 6! = 720 3. How many permutations of the letters ABCDEF contain the letters DEF together in any order? If DEF is considered as one letter, then we have 4 letters A B C DEF which can be permuted in 4! ways, DEF can be ordered by its letters in 3! ways. Hence, by the multiplication principle, total number of orderings possible = 4!*3! = 24*6 = 144. 4. The professor’s dilemma: how to arrange four books on OS, seven on programming, and three on data structures on a shelf such that books on the same subject must be together? Section 3.4 Permutations and Combinations 4 Thursday, March 11, 2010
- 8. Permutation Examples 1. Ten athletes compete in an Olympic event. Gold, silver and bronze medals are awarded to the ﬁrst three in the event, respectively. How many ways can the awards be presented? Hence, 3 objects from a pool of 10 = P(10,3) = 720 2. How many ways can six people be seated on six chairs? P(6,6) = 6! = 720 3. How many permutations of the letters ABCDEF contain the letters DEF together in any order? If DEF is considered as one letter, then we have 4 letters A B C DEF which can be permuted in 4! ways, DEF can be ordered by its letters in 3! ways. Hence, by the multiplication principle, total number of orderings possible = 4!*3! = 24*6 = 144. 4. The professor’s dilemma: how to arrange four books on OS, seven on programming, and three on data structures on a shelf such that books on the same subject must be together? (4!*7!*3*)*3! = 24*5040*6*6 = 4,354,560 Section 3.4 Permutations and Combinations 4 Thursday, March 11, 2010
- 9. Combinations ● When order in permutations becomes immaterial, i.e. we are just interested in selecting r objects from n distinct objects, we talk of combinations denoted by C(n,r) or nCr ● For each combination, there are r! ways of ordering those r chosen objects ● Hence, from multiplication principle, ● C(n,r)* r! = P(n,r) ⇒ ■ Note: C(n,r) is much smaller than P(n,r) as seen from the graphs below: Section 3.4 Permutations and Combinations 5 Thursday, March 11, 2010
- 10. Combinations: Special Cases ● C(n,0) = 1 ● Only one way to choose 0 objects from n objects- chose the empty set ● C(n,1) = n ● Obvious, since n ways to choose one object from n objects ● C(n,n) = 1 ● Only one way to choose n objects from n objects Section 3.4 Permutations and Combinations 6 Thursday, March 11, 2010
- 11. Combinations: Examples ● How many ways can we select a committee of three from 10? ● How many ways can a committee of two women and three men be selected from a group of ﬁve different women and six different men? ● How many ﬁve-card poker hands can be dealt from a standard 52-card deck? ● How many poker hands contain cards all of the same suit? ● How many poker hands contain three cards of one denomination and two cards of another denomination? Section 3.4 Permutations and Combinations 7 Thursday, March 11, 2010
- 12. Combinations: Examples ● How many ways can we select a committee of three from 10? C(10,3) = 120 ● How many ways can a committee of two women and three men be selected from a group of ﬁve different women and six different men? ● How many ﬁve-card poker hands can be dealt from a standard 52-card deck? ● How many poker hands contain cards all of the same suit? ● How many poker hands contain three cards of one denomination and two cards of another denomination? Section 3.4 Permutations and Combinations 7 Thursday, March 11, 2010
- 13. Combinations: Examples ● How many ways can we select a committee of three from 10? C(10,3) = 120 ● How many ways can a committee of two women and three men be selected from a group of ﬁve different women and six different men? For selecting two out of five women, we have C(5,2) ways = 10. For selecting three out of six men, we have C(6,3) ways = 20. Total number of ways for selecting the committee = 10*20 = 200 ● How many ﬁve-card poker hands can be dealt from a standard 52-card deck? ● How many poker hands contain cards all of the same suit? ● How many poker hands contain three cards of one denomination and two cards of another denomination? Section 3.4 Permutations and Combinations 7 Thursday, March 11, 2010
- 14. Combinations: Examples ● How many ways can we select a committee of three from 10? C(10,3) = 120 ● How many ways can a committee of two women and three men be selected from a group of ﬁve different women and six different men? For selecting two out of five women, we have C(5,2) ways = 10. For selecting three out of six men, we have C(6,3) ways = 20. Total number of ways for selecting the committee = 10*20 = 200 ● How many ﬁve-card poker hands can be dealt from a standard 52-card deck? C(52,5) = 2,598,960 ● How many poker hands contain cards all of the same suit? ● How many poker hands contain three cards of one denomination and two cards of another denomination? Section 3.4 Permutations and Combinations 7 Thursday, March 11, 2010
- 15. Combinations: Examples ● How many ways can we select a committee of three from 10? C(10,3) = 120 ● How many ways can a committee of two women and three men be selected from a group of ﬁve different women and six different men? For selecting two out of five women, we have C(5,2) ways = 10. For selecting three out of six men, we have C(6,3) ways = 20. Total number of ways for selecting the committee = 10*20 = 200 ● How many ﬁve-card poker hands can be dealt from a standard 52-card deck? C(52,5) = 2,598,960 ● How many poker hands contain cards all of the same suit? 4*C(13,5) = 5148 ● How many poker hands contain three cards of one denomination and two cards of another denomination? Section 3.4 Permutations and Combinations 7 Thursday, March 11, 2010
- 16. Combinations: Examples ● How many ways can we select a committee of three from 10? C(10,3) = 120 ● How many ways can a committee of two women and three men be selected from a group of ﬁve different women and six different men? For selecting two out of five women, we have C(5,2) ways = 10. For selecting three out of six men, we have C(6,3) ways = 20. Total number of ways for selecting the committee = 10*20 = 200 ● How many ﬁve-card poker hands can be dealt from a standard 52-card deck? C(52,5) = 2,598,960 ● How many poker hands contain cards all of the same suit? 4*C(13,5) = 5148 ● How many poker hands contain three cards of one denomination and two cards of another denomination? Order of events: Select first denomination, select three cards from this denomination, select the second denomination, select two cards from this denomination 13*C(4,3)*12*C(4,2) = 3744. Section 3.4 Permutations and Combinations 7 Thursday, March 11, 2010
- 17. Combinations: Examples ● How many routes are there from the lower-left corner of an n by n square grid to the upper-right corner if we are restricted to traveling only to the right (R) or upward (U)? Consider a 4×4 grid. Total steps required to get from A to B is 8. This can be a mixture of R’s and U’s as shown by the two paths in green and red above. So, 4 R’s and 4 U’s are required but the order is in which step is taken when is not important. ● In how many ways can three athletes be declared winners from a group of 10 athletes who compete in an Olympic event? Section 3.4 Permutations and Combinations 8 Thursday, March 11, 2010
- 18. Combinations: Examples ● How many routes are there from the lower-left corner of an n by n square grid to the upper-right corner if we are restricted to traveling only to the right (R) or upward (U)? Consider a 4×4 grid. Total steps required to get from A to B is 8. This can be a mixture of R’s and U’s as shown by the two paths in green and red above. So, 4 R’s and 4 U’s are required but the order is in which step is taken when is not important. So, basically, we are selecting four objects from eight that can be done in C(8,4) ways. For an n×n grid, one can form C(2n,n) such routes to go from lower-left to the upper-right corner. ● In how many ways can three athletes be declared winners from a group of 10 athletes who compete in an Olympic event? Section 3.4 Permutations and Combinations 8 Thursday, March 11, 2010
- 19. Combinations: Examples ● How many routes are there from the lower-left corner of an n by n square grid to the upper-right corner if we are restricted to traveling only to the right (R) or upward (U)? Consider a 4×4 grid. Total steps required to get from A to B is 8. This can be a mixture of R’s and U’s as shown by the two paths in green and red above. So, 4 R’s and 4 U’s are required but the order is in which step is taken when is not important. So, basically, we are selecting four objects from eight that can be done in C(8,4) ways. For an n×n grid, one can form C(2n,n) such routes to go from lower-left to the upper-right corner. ● In how many ways can three athletes be declared winners from a group of 10 athletes who compete in an Olympic event? C(10,3) = 120 (much less than to award three winners medals) Section 3.4 Permutations and Combinations 8 Thursday, March 11, 2010
- 20. Eliminating Duplicates ● How many ways can a committee of two be chosen from four men and three women and it must include at least one man. ● How many distinct permutations can be made from the characters in the word FLORIDA? ● How many distinct permutations can be made from the characters in the word MISSISSIPPI? Section 3.4 Permutations and Combinations 9 Thursday, March 11, 2010
- 21. Eliminating Duplicates ● How many ways can a committee of two be chosen from four men and three women and it must include at least one man. ● How many distinct permutations can be made from the characters in the word FLORIDA? ● How many distinct permutations can be made from the characters in the word MISSISSIPPI? Section 3.4 Permutations and Combinations 9 Thursday, March 11, 2010
- 22. Eliminating Duplicates ● How many ways can a committee of two be chosen from four men and three women and it must include at least one man. Incorrect and impulsive answer = C(4,1)*C(6,1) Correct answer = C(7,2) – C(3,2) = C(4,1)*C(6,1) – C(4,2) C(4,2) is the number of committees with two men on it. It has to be subtracted since we are counting it twice in C(4,1)*C(6,1). C(7,2) = all committees possible C(3,2) = all committees with no men on it ● How many distinct permutations can be made from the characters in the word FLORIDA? ● How many distinct permutations can be made from the characters in the word MISSISSIPPI? Section 3.4 Permutations and Combinations 9 Thursday, March 11, 2010
- 23. Eliminating Duplicates ● How many ways can a committee of two be chosen from four men and three women and it must include at least one man. Incorrect and impulsive answer = C(4,1)*C(6,1) Correct answer = C(7,2) – C(3,2) = C(4,1)*C(6,1) – C(4,2) C(4,2) is the number of committees with two men on it. It has to be subtracted since we are counting it twice in C(4,1)*C(6,1). C(7,2) = all committees possible C(3,2) = all committees with no men on it ● How many distinct permutations can be made from the characters in the word FLORIDA? Simple: 7! ● How many distinct permutations can be made from the characters in the word MISSISSIPPI? Section 3.4 Permutations and Combinations 9 Thursday, March 11, 2010
- 24. Eliminating Duplicates ● How many ways can a committee of two be chosen from four men and three women and it must include at least one man. Incorrect and impulsive answer = C(4,1)*C(6,1) Correct answer = C(7,2) – C(3,2) = C(4,1)*C(6,1) – C(4,2) C(4,2) is the number of committees with two men on it. It has to be subtracted since we are counting it twice in C(4,1)*C(6,1). C(7,2) = all committees possible C(3,2) = all committees with no men on it ● How many distinct permutations can be made from the characters in the word FLORIDA? Simple: 7! ● How many distinct permutations can be made from the characters in the word MISSISSIPPI? Since we have more than one S, interchanging the S’s at the same position will not result in a distinguishable change. Hence for four S’s, 4! possible permutations that look alike. Hence total number of permutations = Section 3.4 Permutations and Combinations 9 Thursday, March 11, 2010
- 25. Summary of Counting Techniques You want to count the number of… Technique to try: Subsets of an n-element set Use formula 2n Outcomes of successive events Multiply the number of outcomes for each event. Outcomes of disjoint events Add the number of outcomes for each event. Outcomes of speciﬁc choices at each step Draw a decision tree and count the number of paths. Elements in overlapping sections of related Use principle of inclusion and exclusion sets formula Ordered arrangements of r out of n distinct Use P(n,r) formula objects Ways to select r out of n distinct objects Use C(n,r) formula Ways to select r out of n distinct objects Use C(r+n-1, r) formula with repetition allowed Section 3.4 Permutations and Combinations 10 Thursday, March 11, 2010 Table 3.2, page 241 in your text.
- 26. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? ● How many distinct permutations of the characters in ERROR are there? ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. ● Find the number of sets which has two dimes and two quarters. ● Find the number of sets composed of all dimes or all quarters. Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 27. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? 8! 3.7! ● How many distinct permutations of the characters in ERROR are there? ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. ● Find the number of sets which has two dimes and two quarters. ● Find the number of sets composed of all dimes or all quarters. Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 28. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? 8! 3.7! ● How many distinct permutations of the characters in ERROR are there? 5!/3! ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. ● Find the number of sets which has two dimes and two quarters. ● Find the number of sets composed of all dimes or all quarters. Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 29. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? 8! 3.7! ● How many distinct permutations of the characters in ERROR are there? 5!/3! ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? 11!*C(12,8)*8! ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. ● Find the number of sets which has two dimes and two quarters. ● Find the number of sets composed of all dimes or all quarters. Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 30. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? 8! 3.7! ● How many distinct permutations of the characters in ERROR are there? 5!/3! ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? 11!*C(12,8)*8! ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. C(5,2)*C(7,2) = 10*21 = 210 ● Find the number of sets which has two dimes and two quarters. ● Find the number of sets composed of all dimes or all quarters. Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 31. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? 8! 3.7! ● How many distinct permutations of the characters in ERROR are there? 5!/3! ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? 11!*C(12,8)*8! ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. C(5,2)*C(7,2) = 10*21 = 210 ● Find the number of sets which has two dimes and two quarters. C(5,4) + C(7,4) = 5 + 35 = 40 ● Find the number of sets composed of all dimes or all quarters. Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 32. Class Exercises ● How many permutations of the characters in the word COMPUTER are there? How many of these end in a vowel? 8! 3.7! ● How many distinct permutations of the characters in ERROR are there? 5!/3! ● In how many ways can you seat 11 men and eight women in a row if no two women are to sit together? 11!*C(12,8)*8! ● A set of four coins is selected from a box containing ﬁve dimes and seven quarters. C(5,2)*C(7,2) = 10*21 = 210 ● Find the number of sets which has two dimes and two quarters. C(5,4) + C(7,4) = 5 + 35 = 40 ● Find the number of sets composed of all dimes or all quarters. C(7,3)*C(5,1) + C(7,4) = 210 Section 3.4 Permutations and Combinations 11 Thursday, March 11, 2010
- 33. Set, Combinatorics, Probability, and Number Theory Mathematical Structures for Computer Science Chapter 3 Section 3.6 Binomial Theorem Thursday, March 11, 2010
- 34. Pascal’s Triangle ● Consider the following expressions for the binomials: (a + b)0 = 1 (a + b)1 = a + b (a + b)2 = a2 + 2ab + b2 (a + b)3 = a3 + 3a2b + 3ab2 + b3 (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4 ● Row n of the triangle (n ≥ 0) consists of all the values C(n, r) for 0 ≤ r ≤ n. Thus, the Pascal’s looks like this: Row C(0,0) 0 C(1,0) C(1,1) 1 C(2,0) C(2,1) C(2,2) 2 C(3,0) C(3,1) C(3,2) C(3,3) 3 C(4,0) C(4,1) C(4,2) C(4,3) C(4,4) 4 C(5,0) C(5,1) C(5,2) C(5,3) C(5,4) C(5,5) 5 C(n,0) C(n,1) ….………… C(n,n−1) C(n,n) n Section 3.6 Binomial Theorem 13 Thursday, March 11, 2010
- 35. Pascal’s Triangle ● The Pascal’s triangle can be written as: C(0,0) C(1,0) C(1,1) C(2,0) C(2,1) C(2,2) C(3,0) C(3,1) C(3,2) C(3,3) C(4,0) C(4,1) C(4,2) C(4,3) C(4,4) C(5,0) C(5,1) C(5,2) C(5,3) C(5,4) C(5,5) ...... ………. C(n,0) C(n,1) ………. C(n,n − 1) C(n,n) ● We note that C(n,k) = C(n − 1,k − 1) + C(n − 1,k) for 1≤ k ≤ n − 1 ● Can be proved simply by expanding the right hand side and simplifying. Section 3.6 Binomial Theorem 14 Thursday, March 11, 2010
- 36. Pascal’s Triangle Coefﬁcients Power 0 0 1 1 1 1 2 1 2 1 3 3 1 3 1 4 6 4 1 4 1 5 10 10 5 1 5 Section 3.6 Binomial Theorem 15 Thursday, March 11, 2010
- 37. Binomial Theorem ● The binomial theorem provides us with a formula for the expansion of (a + b)n. ● It states that for every nonnegative integer n: ● The terms C(n,k) in the above series is called the binomial coefﬁcient. ● Binomial theorem can be proved using mathematical induction. Section 3.6 Binomial Theorem 16 Thursday, March 11, 2010
- 38. Exercises ● Expand (x + 1)5 using the binomial theorem C(5,0)x5 + C(5,1)x4 + C(5,2)x3 + C(5,3)x4 + C(5,4)x1 + C(5,0)x0 ● Expand (x − 3)4 using the binomial theorem ● What is the ﬁfth term of (3a + 2b)7 ● What is the coefﬁcient of x5y2z2 in the expansion of (x + y + 2z)9 ? ● Use binomial theorem to prove that: C(n,0) – C(n,1) + C(n,2) - …….+ (-1)nC(n,n) = 0 Section 3.6 Binomial Theorem 17 Thursday, March 11, 2010

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