Possibility Trees - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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Counting Techniques:

Possibility Trees,

Multiplication Rule,

Permutations

Possibility Trees

Start

A B

A (A wins) B^ A^

B (B wins)

A (A wins)

B (B wins)

A (A wins)

B (B wins)

Winner of set 1

Winner of set 2

Winner of set 3

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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Possibility trees and Multiplication Rule

Start

B1 B2 B

K1 K2 K1 K2 K1 K

P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P

Select the basic unit Select the keyboard

Example(cont.): The possibility tree:

Select the printer

The number of distinct systems is: 3 ∙2∙2=

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The Multiplication Rule

If an operation consists of k steps and

• the 1st^ step can be performed in n 1 ways,
• the 2nd^ step can be performed in n 2 ways

(regardless of how the 1st^ step was performed) ,

• the k th^ step can be performed in nk ways

(regardless of how the preceding steps were performed) ,

then the entire operation can be performed

in n 1 ∙ n 2 ∙… ∙ nk ways.

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.

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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 ,

10 ∙9∙8∙7 = 5,040 PINs are possible.

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!Docsity.com

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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.