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
• Counting and Probability Problem Solver

• Comments, Questions?

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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 − 2) ... 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