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3. Permutations (Ordered Arrangements)

An arrangement (or ordering) of a set of objects is called a permutation. (We can also arrange just part of the set of objects.)

In a permutation, the order that we arrange the objects in is important

Example

Consider arranging 3 letters: A, B, C. How many ways can this be done?

Answer

Reminder - Factorial Notation

Recall from the Factorial section that n factorial (written n!) is defined as:

n! = n × (n − 1) × (n − 1) ... 3 × 2 × 1

Each of the theorems in this section use factorial notation.

Theorem 1 - Arranging n Objects

In general, n distinct objects can be arranged in n! ways.

Example

In how many ways can 4 different resistors be arranged in series?

Answer

Theorem 2 - Number of Permutations

The number of permutations of n distinct objects taken r at a time, denoted by Pⁿ_r,where repetitions are not allowed, is given by

Notes:

(1) Pⁿ_n = n! (since 0! = 1)

(2) Some books use the following notation for the number of permutations:

nPr

and others have:

ⁿP_r

Example

In how many ways can a supermarket manager display 5 brands of cereals in 3 spaces on a shelf?

Answer

Example

How many different number-plates for cars can be made if each number-plate contains four of the digits 0 to 9 followed by a letter A to Z, assuming that

(a) no repetition of digits is allowed?

(b) repetition of digits is allowed?

Answer

Theorem 3 - Permutations of Different Kinds of Objects

The number of different permutations of n objects of which n₁ are of one kind, n₂ are of a second kind, ... n_k are of a k-th kind is

Example

In how many ways can the six letters of the word "mammal" be arranged in a row?

Answer

Theorem 4 - Arranging Objects in a Circle

There are (n - 1)! ways to arrange n distinct objects in a circle.

Example

In how many ways can 5 people be arranged in a circle?

Answer

Exercises

Exercise 1

In how many ways can 6 girls and 2 boys be arranged in a row

(a) without restriction?

(b) such that the 2 boys are together?

(c) such that the 2 boys are not together?

Answer

Exercise 2

How many numbers greater than 1000 can be formed with the digits 3, 4, 6, 8, 9 if a digit cannot occur more than once in a number?

Answer

Exercise 3

How many different ways can 3 red, 4 yellow and 2 blue bulbs be arranged in a string of Christmas tree lights with 9 sockets?

Answer

Exercise 4

In how many ways can 5 people be arranged in a circle such that two people must sit together?

Answer

 

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Chapter Contents

• Counting and Probability - Introduction
• 1. Factorial Notation
• 2. Basic Principles of Counting
• 3. Permutations
• 4. Combinations
• 5. Introduction to Probability Theory
• 6. Probability of an Event
• Singapore TOTO
• Probability and Poker
• 7. Conditional Probability
• 8. Independent and Dependent Events
• 9. Mutually Exclusive Events
• 10. Bayes’ Theorem
• 11. Probability Distributions - Concepts
• 12. Binomial Probability Distributions
• 13. Poisson Probability Distribution
• 14. Normal Probability Distribution
• The z-Table

Following are the original SNB files (.tex or .rap) used in making this chapter. For more information, go to SNB info.

SNB files

1. Factorial Notation

2. Basic Principles of Counting

3. Permutations

4. Combinations

5. Introduction to Probability Theory

6. Probability of an Event

Singapore TOTO

Probability and Poker

7. Conditional Probability

8. Independent and Dependent Events

9. Mutually Exclusive Events

10. Bayes’ Theorem

11. Probability Distributions - Concepts

12. Binomial Probability Distributions

13. Poisson Probability Distribution

14. Normal Probability Distribution

• Comments, Questions?

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