1. Chapter 10
Counting Methods
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2. Chapter 10: Counting Methods
10.1 Counting by Systematic Listing
10.2 Using the Fundamental Counting Principle
10.3 Using Permutations and Combinations
10.4 Using Pascal’s Triangle
10.5 Counting Problems Involving “Not” and “Or”
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3. Using the Fundamental Counting Principle
Section 10-2
Using the Fundamental Counting Principle
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4. Using the Fundamental Counting Principle
Uniformity and the Fundamental Counting Principle
Factorials
Arrangements of Objects
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5. Uniformity Criterion for Multiple-Part Tasks
A multiple-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 the previous parts.
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6. Fundamental Counting Principle
When a task consists of k separate parts and satisfies the uniformity criterion, if the first part can be done in n1 ways, the second part can be done in n2 ways, and so on through the k th part, which can be done in nk ways, then the total number of ways to complete the task is given by the product
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7. Example: Two-Digit Numbers
How many two-digit numbers can be made from the set {0, 1, 2, 3, 4, 5}? (numbers can’t start with 0.)
Solution
Part of Task
Select first digit
Select second digit
Number of ways 5
(0 can’t be used) 6
There are 5(6) = 30 two-digit numbers.
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8. Example: Two-Digit Numbers with Restrictions
How many two-digit numbers that do not contain repeated digits can be made from the set {0, 1, 2, 3, 4, 5} ?
Solution
Part of Task
Select first digit
Select second digit
Number of ways 5
(repeated digits not allowed)
There are 5(5) = 25 two-digit numbers.
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9. Example: Two-Digit Numbers with Restrictions
How many ways can you select two letters followed by three digits for an ID?
Solution
Part of Task
First letter
Second letter
Digit
Number of ways 26 10
There are 26(26)(10)(10)(10) = 676,000 IDs possible.
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10. Factorials
For any counting number n, the product of all counting numbers from n down through 1 is called n factorial, and is denoted n!.
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11. Factorial Formula
For any counting number n, the quantity n factorial is given by
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12. Example: Solution Evaluate each expression. a) 4! b) (4 – 1)! c)
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13. Definition of Zero Factorial
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14. Arrangements of Objects
When finding the total number of ways to arrange a given number of distinct objects, we can use a factorial.
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15. Arrangements of n Distinct Objects
The total number of different ways to arrange n distinct objects is n!.
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16. Example: Arranging Books
How many ways can you line up 6 different books on a shelf?
Solution
The number of ways to arrange 6 distinct objects is 6! = 720.
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17. Arrangements of n Objects Containing Look-Alikes
The number of distinguishable arrangements of n objects, where one or more subsets consist of look-alikes (say n1 are of one kind, n2 are of another kind, …, and nk are of yet another kind), is given by
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18. Example: Distinguishable Arrangements
Determine the number of distinguishable arrangements of the letters of the word INITIALLY.
Solution
9 letters total
3 I’s and 2 L’s
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