1. Chapter 10: Counting Methods

2. Counting by Systematic Listing
10.1
One-part tasks: each item in the list is identified by one feature
Multi-part tasks: items in the list are identified by more than one feature (ex a playing card has a suit and a value)

3. Two-part tasks: Using Tables
10.1
Two-part tasks: Using Tables
Example: Determine the number of two-digit numbers that can be made using 2,4 and 7 for each digit
Example: Determine how many ways you can pick two people from the group of Andria, Bridget, Connor and Dylan.

4. Uniformity Criterion
10.2
A multi-part task is said to satisfy the uniformity criterion if the number of choices for any particular part is the same no matter which choices were selected for previous parts

5. Fundamental Counting Principle (FCP)
10.2
Fundamental Counting Principle (FCP)
For a task that consists of k separate parts and satisfies the uniformity criterion
Suppose there are n1 ways to do the first task, n2 ways to do the second, etc
Then total number of ways to complete the task is
n1 × n2 × … × nk

6. Example
10.2
Using the digits 0,1,2,3,4,5,6,7,8,9, how many 4 digit numbers are there such that
0 isn’t used for first, repetition is okay
0 isn’t used for first, numbers all distinct
Number is odd and greater than 5000, repetition is okay

7. Factorials and Arrangements
10.2
n! = n × (n-1) × (n-2) × … 2 × 1
0! = 1
The total number of different ways to arrange n distinct objects is n!

8. Permutations
10.3

9. Combinations
10.3

10. Guidelines for Choosing Counting Method
10.3
If selected items can be repeated or are separate, use FCP
If selected items cannot be repeated and order is important, use permutations
If selected items cannot be repeated and order isn’t important, use combinations

11. Pascal’s Triangle
10.4
C(n,r) gives the number in the rth spot of of the nth row

12. Counting and Gambling
In a standard deck of cards, each card has a suit (clubs, spades, hearts or diamonds) and a rank (2,3,4,5,6,7,8,9,10, Jack, Queen, King, Ace).
There are 4×13 =52 cards in total
A poker hand has 5 different cards from the deck

13. Poker Hands
How many hands in total? C(52,5) How many four of a kind? How many straights? How many three of a kind?