PERMUTATION AND COMBINATION (EXERCISE)

EXERCISE

PERMUTATION

      In class of 30 students, a first and second prize are to be awarded. 
     In how many different ways can this be done?

     

          N!        =      30!       = 30 x 29 x 28 x 27(eliminate the same number) = 30 x 29 x 28 = 24360 ways

     ( N - n)!        (30 - 3)!                  27                                                                    

     COMBINATION

                    

     
     In a class of 30, 11 boys and the rest are girls. A student body of 3 prefects are to be elected. 
     How many ways can they be chosen if
     a. Both prefects elected are girls
     b. Both prefects elected are boys
     c. 1 girl 1 boy prefects 

   a) girls:

 

               N!       =     19!    =    19!    =  19 x 18 x 17 x 16  (eliminate the same number)   = 19 x 18 x17

         n!(N - n)!    3!(19-3)!    3 x 16!        3 x 2 x 1 x 16                                                          3 x 2 x 1      

        = 5814  = 969 ways 

              6

      
        b) boys:


         N!       =       11!     =    11!     =      11 x 10 x 9     (eliminate the same number)   =      11       = 11
         n!(N - n)!      3!(11-3)!    3 x 8!      3 x 2 x 1 x 10 x 9                                                     3 x 2 x 1      6

        = 1.8333..3 ways


         c) 1 girl 1 boy:

               N!       =       11!     =    11!     =   11 x 10 x 9 x 8 x7  (eliminate the same number)  

         n!(N - n)!      1!(11-3)!     1 x 8!          3 x 2 x 1 x 8 x 7                                                  

        

       = 11 x 10 x 9   =  990  = 165 ways

            3 x 2 x 1           6

        

PERMUTATION AND COMBINATION

Permutation and Combination, which one is which?

Let's start with Permutation,

The number of different ways that a certain number of objects can be arranged in order 

from a large number of objects.

- Ordered lists (Order matters)

- Key Word : Arrangements

FORMULA

!    =   Factorial (number of ways)

N! =   Permutation

n   =   Number selected

     N !          

( N - n ) ! 

Example:


1! = 1                                                      = 1
2! = 2 x 1                                         = 2
3! = 3 x 2 x 1                                          = 6
4! = 4 x 3 x 2 x 1                             = 24
5! = 5 x 4 x 3 x 2 x 1                       = 120
6! = 6 x 5 x 4 x 3 x 2 x 1                 = 720
7! = 7 x 6 x 5 x 4 x  3 x 2 x 1          = 5,040   




Second, Combination

The number of different ways that a certain number of objects as a group can be selected
from a larger number of objects.

- Unordered group/ set : order does not matter
- Key Word : Choice, Selection, Election

FORMULA

!    =  Factorial (number of ways)
N! =  Combination
n   =  Number selected

        N !       
n ! ( N - n ) !

PROBABILITY (EXAMPLE & EXERCISE)

EXAMPLE


 A coin flipped twice. Draw a tree diagram to show all the possible outcomes.

OUTCOMES PROBABILITIES

HH 1/2 x 1/2 = 1/4

HT 1/2 x 1/2 = 1/4

TH  1/2 x 1/2 = 1/4

TT  1/2 x 1/2 = 1/4

EXERCISE

a) What is the probability of getting blue twice?

BB = 1/5 x 1/5  =  1/25

b) What is the probability of not getting blue twice?

BR = 1/5 x 4/5  =  4/25

RB = 4/5 x 1/5  =  4/25

RR = 4/5 x 4/5  =  16/25

4/25 + 4/25 + 16/25  =  24/25

c) What is the probability of getting the same color twice?

BB = 1/5 x 1/5 = 1/25

RR = 4/5 x 4/5 = 16/25

1/25 + 16/25 = 17/25

d) What is the probability of getting different colors?

BR = 1/5 x 4/5 = 4/25

RB = 4/5 x 1/5 = 4/25

4/25 + 4/25 = 8/25

PROBABILITY

Probability is the measure of how likely an event is

Experiment is a situation involving chance or probability that leads to results called outcome

Outcome / Sample point is the result of a single trial of an experiment

Sample space / Event all possible outcomes of an experiment (s)

In order to measure probabilities, mathematicians have devised the following formula for finding the probability of an event :

FORMULA

P(A) = the number of ways 'A' can occur
            total number of possible outcome

P ; Probability
A ; Event

0 = will not be less than 0, (impossible)
1 = will not be more than 1, (certain)

The word AND in probability means the intersection of 2 events

AND = n
OR = u

P( E n F)

E = rolling an even number
F = rolling a number greater than 3


The word OR in probability means the union of 2 events

P(E u F)


ADDITION RULE

- For any two events E and F

P ( E u F ) = P(E) + P(F) - P( E n F)

ADDITION RULE FOR MUTUALLY EXCLUSIVE EVENTS

if E and F are mutually exclusive events,

P( E u F) = P( E ) + P( F )

MEASURE OF CENTRAL TENDENCY

LEARNING OBJECTIVES

- To be able to find the mean, median and mode from a set of data

- To be able to find the range of a set of data

- To be able to find the mean from continuous grouped data

- To be able to interpret and compare data sets

KEY WORDS

- Mean

- Median

- Mode

- Range

- Frequency distribution

AVERAGES

Averages are used to summarise what the data shows. 

There are 3 types of average: mean, median, mode

The range of distribution can also be found

Lets start with Mode

The mode is the most common value or a repeatative value in a set of data

Example, 

Find the mode of red, blue, green, red, black, red, yellow, green 

Mode = Red, Green

It is also possible to have no mode at all

Example: 1, 2, 3, 4, 5, 6, 7.....

Median

The median is the middle value in a set of data

The first step is to put the numbers in numerical order

Example:

Find the median of 4, 6, 7, 2, 9, 1, 3, 5, 8

Ordered : 1, 2, 3, 4, 5, 6, 7, 8, 9

Median = 5

If there are 2 numbers in the middle. The median is in the middle of these because there can only be one median. 

Example:

Find the median of 7, 2, 6, 4, 8, 3

Ordered: 2, 3, 4, 6, 7, 8

Median = 5

Mean 

To find the mean of a set of data

Add all the values together

Divide by the number of values there are in a set

The mean takes the total of all the values and spreads the total out evenly to get an average

Range

The range of a set of data is the difference between the largest and smallest value

Formula :

Range = largest value - smallest value

Analysing large sets of data?

You use a frequency table! It looks like this

Use a tally to create a frequency table from the used to find the mean in a more

REPRESENTATION OF STATISTICAL DATA (EXERCISE)

Group Data

1) The marks of 30 students in an exam are marked out 50

22  32  29  7    13  41  34  28  27
39  18  33  45  28  39  31  17  41
35  28  15  8    33  47  21  27  34
36  33  29

2)

3)

REPRESENTATION OF STATISTICAL DATA

Learning Objectives:

- To be able to collect data in tally table
- To recognise the difference between discrete and continous data
- To be able to represent data using a bar chart, frequency diagram, frequency polygon and histogram


Key Words

• Data - Frequency polygon
• Discrete - Histogram
• Continous - Frequency
• Tally - Qualitative
• Bar Chart - Quantitative
• Frequency Diagram 

Data is another word for information. When data is collected there are lots of ways to represent it using different charts, tables and statistics.

There are different types of data:

Quantitative and Qualitative

Quantitative is Numerical data (e.g cost of shirt)

Qualitative is Non-numerical data (e.g the colour of a shirt)

Discrete and Continous 

Discrete data can be counted. They can take particular values (e.g number of children)

Continuous data results when measuring things like length, time and mass. (e.g the time taken to run is 100m)

1) Raw Data, it is not organised in any way,

3 4 5 4 2 4 6 

6 4 1 2 3 4 5

8 7 8 1 3 4 4

5 6 6 5 4 2 5

To begin to analyse the data is to organise it into a 2) Tally chart

Each tally mark represents one piece of data

Frequency gives the total count of each size

Drawing a 3) Bar chart

The data on shoe sizes is discrete data. Therefore, you can draw a bar chart.

Frequency Diagram 

Another graph can be created by joining the midpoints

DRAWING HISTOGRAMS

Each group or class is represented by a bar. There are no gaps between the bars. 

The area of each bar is proportional to the frequency of the class it represents.

FORMULA:

Frequency density = frequency

                                 class width

The frequency density is calculated for each class and gives the height of each bar. 

The vertical axis of the histogram is labelled "frequency density"

It is similar to a frequency diagram, only the vertical axis is frequency density instead of frequency