Combinatorics - Elementary Discrete Math - Lecture Slides, Slides of Discrete Mathematics

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Elementary Counting

Technique s & Combinatorics

Consider:

• How many license plates are possible with 3 letters followed by 3 digits?
• How many license plates are possible with 3 letters followed by 3 digits if no letter repeated?
• How many different ways can we chose from 4 colors and paint 6 rooms?

Combinatorics

• the branch of discrete mathematics concerned with determining the size of finite sets without actually enumerating each element.

Combinatorics

• The Sum Rule (task formulation):
  • Suppose that a task can be completed by performing exactly one task from a collection of disjoint subtasks: subtask 1 , subtask 2 , ... , subtask n ;
  • Now suppose each subtask has a choice of ways to perform it, e.g. - subtask 1 can be performed t 1 ways, - subtask 2 can be performed t 2 ways, - ... - subtask (^) n can be performed t n ways.
  • Then number the number of ways to perform the task is: t 1 + t 2 + ... + t n

The Sum Rule

• Example:
  • Suppose either a CS faculty or a CS student must be chosen for a committee, and there are 4 CS faculty and 16 CS students. How many possible choices are there?

The Sum Rule

• Example:
  • Suppose a student can meet the humanities course requirement by taking either a religion, literature, or art course. There are 3 religion, 4 literature, or 4 art courses to chose from. How many possible choices are there?

Example - Product Rule

• How many different ways can we chose from 4 colors and paint 3 rooms? - Tasks: - 1 - paint room 1 - 4 ways to perform (4 colors) - 2 - paint room 2 - 4 ways to perform (4 colors) - 3 - paint room 3 - 4 ways to perform (4 colors) - Thus t 1 = 4, t 2 = 4, t 3 = 4, and 4 ⋅ 4 ⋅ 4 = 64 ways to paint the rooms

Example - Product Rule

• How many different ways can we chose from 4 colors and paint 3 rooms, if no room is to be the same color? - tasks: - 1 - paint room 1 - 4 ways to perform (4 colors) - 2 - paint room 2 - 3 ways to perform (3 colors left) - 3 - paint room 3 - 2 ways to perform (2 colors left) - Thus t 1 = 4, t 2 = 3, t 3 = 2, and 4 ⋅ 3 ⋅ 2 = 24 ways to paint the rooms

Example - Product Rule

• How many different 3 people can be selected from a group of 8 people to a president, vice- president, treasure of the group?

Example - Product Rule

• If student ID’s are two capital letters followed by three numeric digits, then how many ID’s are possible? - What if the two letters must be distinct? - What if the letters and the numbers must all be distinct?

Example - Product Rule

Let A = {a, b, c, d, e}, B = {1, 3, 5, 7}

• How many pairs ( x, y ) exist where x ∈ A and y ∈ B?
• A × B has cardinality |A| ⋅ |B| = 5 ⋅ 4 = 20

Example - Product Rule

• How many license plates are possible with 3 letters followed by 3 digits?

Example - The Sum Rule

Let A = {a, b, c, d, e}, B = {1, 3, 5, 7}

• How many ways can one element be chosen?

|A ∪ B| = |A| + |B| = 5 + 4 = 9.

Example - The Sum Rule

• You have five novels, four magazines, and three devotional books. - How many options do you have for taking one for your wait in the bank line?