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Equally Likely Outcomes Permutations and Combinations Examples Counting Techniques Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College of Engineering and Science Counting Techniques Equally Likely Outcomes

  1. Equally Likely Outcomes Permutations and Combinations Examples Definition. The outcomes in an event A for which we want to compute the probability are also called favorable outcomes . Theorem. Let S be a sample space with a probability function P so that every individual outcome/element in S has the same probability. Then the probability of an event A is equal to the number of elements in A divided by the number of elements in S . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  2. Equally Likely Outcomes Permutations and Combinations Examples Definition. The outcomes in an event A for which we want to compute the probability are also called favorable outcomes . Theorem. Let S be a sample space with a probability function P so that every individual outcome/element in S has the same probability. Then the probability of an event A is equal to the number of elements in A divided by the number of elements in S . In other words, when all outcomes are equally likely, the probability of an event is the number of favorable outcomes divided by the total number of outcomes. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  3. Equally Likely Outcomes Permutations and Combinations Examples Definition. The outcomes in an event A for which we want to compute the probability are also called favorable outcomes . Theorem. Let S be a sample space with a probability function P so that every individual outcome/element in S has the same probability. Then the probability of an event A is equal to the number of elements in A divided by the number of elements in S . In other words, when all outcomes are equally likely, the probability of an event is the number of favorable outcomes divided by the total number of outcomes. P ( A ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  4. Equally Likely Outcomes Permutations and Combinations Examples Definition. The outcomes in an event A for which we want to compute the probability are also called favorable outcomes . Theorem. Let S be a sample space with a probability function P so that every individual outcome/element in S has the same probability. Then the probability of an event A is equal to the number of elements in A divided by the number of elements in S . In other words, when all outcomes are equally likely, the probability of an event is the number of favorable outcomes divided by the total number of outcomes. P ( A ) = | A | | S | logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  5. Equally Likely Outcomes Permutations and Combinations Examples Definition. The outcomes in an event A for which we want to compute the probability are also called favorable outcomes . Theorem. Let S be a sample space with a probability function P so that every individual outcome/element in S has the same probability. Then the probability of an event A is equal to the number of elements in A divided by the number of elements in S . In other words, when all outcomes are equally likely, the probability of an event is the number of favorable outcomes divided by the total number of outcomes. P ( A ) = | A | | S | = number of favorable outcomes total number of outcomes logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  6. Equally Likely Outcomes Permutations and Combinations Examples Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  7. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  8. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  9. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  10. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? We are looking at triples (appetizer, main course, dessert). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  11. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? We are looking at triples (appetizer, main course, dessert). There are logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  12. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? We are looking at triples (appetizer, main course, dessert). There are 6 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  13. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? We are looking at triples (appetizer, main course, dessert). There are 6 · 4 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  14. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? We are looking at triples (appetizer, main course, dessert). There are 6 · 4 · 10 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  15. Equally Likely Outcomes Permutations and Combinations Examples Definition. To keep track of outcomes that happen in a certain order, we can consider ordered k-tuples of elements ( x 1 ,..., x k ) . Example. How many meals can be composed if there are 6 choices for appetizers, 4 choices for the main course and 10 choices for desserts? We are looking at triples (appetizer, main course, dessert). There are 6 · 4 · 10 = 240 of them. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  16. Equally Likely Outcomes Permutations and Combinations Examples Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  17. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  18. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  19. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  20. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  21. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  22. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  23. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  24. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  25. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  26. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  27. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  28. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  29. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  30. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  31. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  32. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of 52 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  33. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of 52 · 51 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  34. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of 52 · 51 · 50 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  35. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of 52 · 51 · 50 · 49 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  36. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of 52 · 51 · 50 · 49 · 48 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  37. Equally Likely Outcomes Permutations and Combinations Examples Theorem. If there are n 1 ways to choose the first object, n 2 ways to choose the second, etc. and n k ways to choose the k th object, then there are n 1 · n 2 ··· n k ordered k-tuples. Example. When 5 cards are dealt in a poker hand, the deal can be modeled as an ordered 5 -tuple of cards: (first card, second card, third card, fourth card, fifth card). If we consider a deal in which all cards are given to you right away (like in a video poker machine), then there are 52 possibilities for the first card, 51 possibilities for the second card, 50 possibilities for the third card, 49 possibilities for the fourth card, 48 possibilities for the fifth card, for a total of 52 · 51 · 50 · 49 · 48 = 311 , 875 , 200 possible ways the deal could happen. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  38. Equally Likely Outcomes Permutations and Combinations Examples Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  39. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  40. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  41. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  42. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  43. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  44. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  45. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . We also define 0! = 1 . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  46. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . We also define 0! = 1 . (This makes certain formulas consistently applicable for “borderline cases”.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  47. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . We also define 0! = 1 . (This makes certain formulas consistently applicable for “borderline cases”.) Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  48. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . We also define 0! = 1 . (This makes certain formulas consistently applicable for “borderline cases”.) Theorem. P k , n logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  49. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . We also define 0! = 1 . (This makes certain formulas consistently applicable for “borderline cases”.) Theorem. P k , n = n · ( n − 1 ) · ( n − 2 ) ··· ( n − k + 1 ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  50. Equally Likely Outcomes Permutations and Combinations Examples Definition. An ordered sequence of k objects out of n distinct objects is called a permutation of size k of n objects. The number of permutations of size k of n objects is denoted P k , n . Definition. For any nonnegative integer m, we define the factorial to be m ! : = m · ( m − 1 ) · ( m − 2 ) ··· 3 · 2 · 1 . We also define 0! = 1 . (This makes certain formulas consistently applicable for “borderline cases”.) n ! Theorem. P k , n = n · ( n − 1 ) · ( n − 2 ) ··· ( n − k + 1 ) = ( n − k ) ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  51. Equally Likely Outcomes Permutations and Combinations Examples Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  52. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  53. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  54. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  55. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  56. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  57. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  58. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . The number of combinations is denoted � n � = C k , n k logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  59. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . The number of combinations is denoted � n � = C k , n and called the binomial coefficient , pronounced k “n choose k”. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  60. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . The number of combinations is denoted � n � = C k , n and called the binomial coefficient , pronounced k “n choose k”. Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  61. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . The number of combinations is denoted � n � = C k , n and called the binomial coefficient , pronounced k “n choose k”. � n � Theorem. k logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  62. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . The number of combinations is denoted � n � = C k , n and called the binomial coefficient , pronounced k “n choose k”. � n � = P k , n Theorem. k k ! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  63. Equally Likely Outcomes Permutations and Combinations Examples Example. There are P 5 , 52 = 52! 47! ways to deal a 5 card hand. But this is not the number of different 5 card hands, because for a poker hand, the order of the deal does not matter. As long as order matters, every combination that we commonly consider a “hand” is counted 5! times. Definition. An unordered subset of k objects out of n is called a combination . The number of combinations is denoted � n � = C k , n and called the binomial coefficient , pronounced k “n choose k”. � n � = P k , n n ! Theorem. k ! = k ! ( n − k ) ! . k logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  64. Equally Likely Outcomes Permutations and Combinations Examples Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  65. Equally Likely Outcomes Permutations and Combinations Examples Example. The number of possible 5 card hands out of a 52 � 52 � card deck is 5 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  66. Equally Likely Outcomes Permutations and Combinations Examples Example. The number of possible 5 card hands out of a 52 � 52 � = 52! card deck is 5 5!47! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  67. Equally Likely Outcomes Permutations and Combinations Examples Example. The number of possible 5 card hands out of a 52 � 52 � = 52! 5!47! = 2 , 598 , 960 . card deck is 5 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  68. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  69. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  70. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  71. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  72. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  73. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  74. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? 1. Each suit has 13 cards. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  75. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? 1. Each suit has 13 cards. � 13 � 2. If all cards come from the same suit, then we have 5 ways to get all 5 cards from that suit. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

  76. Equally Likely Outcomes Permutations and Combinations Examples How Many 5 Card Hands are Made up Entirely of Cards in the Same Suit (Flushes)? 1. Each suit has 13 cards. � 13 � 2. If all cards come from the same suit, then we have 5 ways to get all 5 cards from that suit. 3. There are 4 suits. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Counting Techniques

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