Counting Techniques: Possibility Trees, Multiplication Rule, Permutations

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Counting TechniquesPossibility Trees,
Multiplication Rule, Permutations
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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
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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?

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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 32212
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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.

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Multiplication Rule (Example)

• Consider the following nested loop
• for i1 to 5
• for j1 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)

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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,
• 10101010 10,000 PINs are possible.

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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,
• 10987 5,040 PINs are possible.

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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,
• 4321 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.

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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 n1,
• 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)21
  n!

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Example on PermutationsThe 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.