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Counting Techniques: Possibility Trees, Multiplication Rule, Permutations

Counting Techniques: Possibility Trees, Multiplication Rule, Permutations. Possibility Trees. In a tennis match, the first player to win two sets, wins the game. Question: What is the probability that player A will win the game in 3 sets? Construct possibility tree :.

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Counting Techniques: Possibility Trees, Multiplication Rule, Permutations

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  1. Counting Techniques:Possibility Trees, Multiplication Rule, Permutations

  2. Possibility Trees • In a tennis match, the first player to win two sets, wins the game. • Question: What is the probability that player A will win • the game in 3 sets? • Construct possibility tree: Winner of set 1 Winner of set 2 Winner of set 3

  3. Possibility trees and Multiplication Rule • Example: When buying a PC system, you have the choice of  3 models of the basic unit: B1, B2, B3 ;  2 models of keyboard: K1, K2 ;  2 models of printer: P1, P2 . • Question: How many distinct systems can be purchased?

  4. Possibility trees and Multiplication Rule Example(cont.): The possibility tree: Select the basic unit Select the keyboard Select the printer The number of distinct systems is:3∙2∙2=12

  5. The Multiplication Rule If an operation consists of k steps and  the 1st step can be performed in n1 ways,  the 2nd step can be performed in n2 ways (regardless of how the 1st step was performed) , ….  the kth step can be performed in nk ways (regardless of how the preceding steps were performed) , then the entire operation can be performed in n1 ∙ n2 ∙… ∙ nk ways.

  6. Multiplication Rule (Example) • Consider the following nested loop: for i:=1 to 5 for j:=1 to 6 [ Statement 1 ; Statement 2 . ] next j next i • Question: How many times the statements in the inner loop will be executed? • Solution:5 ∙ 6 = 30 times (based on the multiplication rule)

  7. Multiplication Rule (Example) • A PIN is a sequence of any 4 digits (repetitions allowed); e.g., 5279, 4346, 0270. • Question. How many different PINs are possible? • Solution. Choosing a PIN is a 4-step operation:  Step 1: Choose the 1st symbol (10 different ways).  Step 2: Choose the 2nd symbol (10 different ways).  Step 3: Choose the 3rd symbol (10 different ways).  Step 4: Choose the 4th symbol (10 different ways). Based on the multiplication rule, 10∙10∙10∙10 = 10,000 PINs are possible.

  8. Multiplication Rule (Example) • Consider the problem of choosing PINs but now repetitions are not allowed. • Question. How many different PINs are possible? • Solution. Choosing a PIN is a 4-step operation:  Step 1: Choose the 1st symbol (10 different ways).  Step 2: Choose the 2nd symbol (9 different ways).  Step 3: Choose the 3rd symbol (8 different ways).  Step 4: Choose the 4th symbol (7 different ways). Based on the multiplication rule, 10∙9∙8∙7 = 5,040 PINs are possible.

  9. Multiplication Rule and Permutations • Consider the problem of choosing PINs again. Now  a PIN is a sequence of 1, 2, 3, 4 ;  repetitions are not allowed. • Question. How many different PINs are possible? • Solution. Choosing a PIN is a 4-step operation:  Step 1: Choose the 1st symbol (4 different ways).  Step 2: Choose the 2nd symbol (3 different ways).  Step 3: Choose the 3rd symbol (2 different ways).  Step 4: Choose the 4th symbol (1 way). Based on the multiplication rule, 4∙3∙2∙1 = 4! = 24 PINs are possible. • Note:The number of different PINs in this case is just the number of different orders of 1,2,3,4.

  10. Permutations • A permutation of a set of objects is an ordering of the objects in a row. • Example: The permutations of {a,b,c}: abc acb bac bca cab cba • Theorem. For any integer n with n≥1, the number of permutations of a set with n elements is n! . • Proof. Forming a permutation is an n-step operation:  Step 1: Choose the 1st element ( n different ways).  Step 2: Choose the 2nd element ( n-1 different ways). …  Step n: Choose the nth element (1 way). Based on the multiplication rule, the number of permutations is n∙(n-1)∙…∙2∙1 = n!

  11. Example on Permutations:The Traveling Salesman Problem (TSP) • There are n cities. The salesman  starts his tour from City 1,  visits each of the cities exactly once,  and returns to City 1. Question: How many different tours are possible? Answer: Each tour corresponds to a permutation of the remaining n-1 cities. Thus, the number of different tours is (n-1)! . Note: The actual goal of TSP is to find a minimum-cost tour.

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