# Counting-Probability - permutations

### Math Is FUN!!

## WE HEAR!!:Teen mothers who live with their parents are less likely to use marijuana than teen moms in other living arrangements. | ## WE HEAR!!:She'll probably take the offer | ## WE HEAR!!:The chance of rain tomorrow is 75% |

## HOW IT IS RELATED TO MATH ??

## Math and Probability

- To decide "how likely" an event is, we need to count the number of times an event could occur and compare it to the total number of possible events.
- Such a comparison is called the
**probability**of the particular event occurring.

The **Mathmetical** theory of counting is known as

**combinatorial analysis**.

## FActorial NotatiOn

## FActorial nOtaTion USe

- A simple way of writing the product of all the positive whole numbers up to a given number.

## What is N?!!

is the product of all the integers from 1 to*n*factorial*n.*is represented with an exclamation mark*n*factorial**:***n!***n!**=n(n-1)(n-2)...(3)(2)(1)__EXAMPLE:__

__NOTE:__

## counting: Principles and rules

## counting techniques

- Handle large masses of statistical data
- Understanding probability.

## counting principle

## Number of Outcomes of an Event

- event
*E*defined as*E*= "season of the year" - We write the "number of outcomes of event
*E*" as*n*(*E*).

So in the example,

*n*(*E*)=4,

since there are 4 seasons in a year.30 days

## counting rule

## Addition Rule

- Let
*E*1 and*E*2 be**mutually exclusive**events:

The number of times event *E* will occur can be given by the expression:

*n*(*E*) = *n*(*E*1) + *n*(*E*2)

where

*n*(*E*) = Number of outcomes of event *E*

*n*(*E*1) = Number of outcomes of event *E*1

*n*(*E*2) = Number of outcomes of event *E*2

## example

- Consider a set of numbers

*S*={−4,−2,1,3,5,6,7,8,9,10}

- Let the events
*E*1,*E*2 and*E*3 be defined as:

*E *= choosing a negative or an odd number from *S*;

*E*1= choosing a negative number from S;

*E*2 = choosing an odd number from S.

- Find
*n*(*E*).

__Answer:__*E*1 and*E*2 are mutually exclusive events*n*(*E*) =*n*(*E*1) +*n*(*E*2)

= 2 + 5

= 7

## multiplication rule

- Two events
*E*1 and*E*2 are to be performed and the events*E*1 and*E*2 are**independent**events. (one does not affect the other's outcome) - Generalization:

*E*1 can result in any one of

*n*(

*E*1) possible outcomes; and for each outcome of the event

*E*1, there are

*n*(

*E*2) possible outcomes of event

*E*2.

Together there will be *n*(*E*1) × *n*(*E*2) possible outcomes of the two events.

That is, if event *E* is the event that both *E*1 and *E*2 **must** occur, then

*n*(*E*) =*n*(*E*1) ×*n*(*E*2)

## example 1

- Say the only clean clothes you've got are

__Answer:__- We have :

2 t-shirts and with each t-shirt we could pick 4 pairs of jeans. Altogether there are

2×4=8 possible combinations.

- We could write

*E*1 = "choose t-shirt" and

*E*2 = "choose jeans"

*n*(*E*1) = 2 (since we had 2 t-shirts)

*n*(*E*2) = 4 (since there were 4 pairs of jeans)

- So total number of possible outcomes is given by:

*n*(*E*) = *n*(*E*1) × *n*(*E*2) = 2 × 4 = 8

## example 2

- What is the total number of possible outcomes when a pair of coins is tossed?

__Answers:__*The events are described as:*

*E*1 = toss first coin (2 outcomes, so *n*(*E*1) = 2.)

*E*2 = toss second coin (2 outcomes, so *n*(*E*2) = 2.)

- They are independent, since neither toss affects the outcome of the other toss.
- So
*n*(*E*) =*n*(*E*1) ×*n*(*E*2) = 2 × 2 = 4

## permutations

## Theorem 1 - Arranging n Objects

*n*distinct objects can be arranged in

*n*! ways.

## Theorem 2 - Number of Permutations

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

*P**n**r*=*n*(*n*−1)(*n*−2)...(*n*−*r*+1)=*n*!(*n*−*r*)!

## Theorem 3 - Permutations of Different Kinds of Objects

The number of different permutations of *n* objects of which *n*1 are of one kind, *n*2 are of a second kind, ... *nk *are of a *k*-th kind is

*n*!*n*1!×*n*2!×*n*3×...×*n**k*!

## Theorem 4 - Arranging Objects in a Circle

*n*−1)! ways to arrange

*n*distinct objects in a circle