Pensil Harapan

Permutation and Combination

Permutation : Permutation means arrangement of things. The word arrangement is used, if the order of things is considered.

Combination: Combination means selection of things. The word selection is used, when the order of things has no importance.

Distinctly ordered sets are called arrangements or permutations.

Example :

A flutter on the horses There are 7 horses in a race.

a) In how many different orders can the horses finish?

b) How many trifectas (1st , 2nd and 3rd) are possible?

Solution :

a) 7.6.5.4.3.2.1=7! OR 7P7

b) 7.6.5 = 210 or 7P3

Permutations with Restrictions

In how many ways can 5 boys and 4 girls be arranged on a bench if there are

a) no restrictions?

b) boys and girls alternate?

c) Anne and Jim wish to stay together?

Solution :

a) 9!

b) A boy will be on each end BGBGBGBGB = 5.4.4. 3.3.2.2.1.1

Or 5! x 4! or 5P5x4P4

c) (AJ) _ _ _ _ _ _ _ = 2.8! or 2.8P8

Arrangements with Repetitions

How many different arrangements of the word PARRAMATTA are possible? Arrangements with Repetitions

P A R R A M

Solution : 10 letters but note repetition (4 A’s, 2 R’s, 2 T’s)

No. of arrangements = 10!/(4! 2! 2! ) = 37 800

Combination

The number of different combinations (i.e. unordered sets)

No.of combination = (number of permutations )/( arrangements of r objects)

Example :

How many ways can a basketball team of 5 players be chosen from 8 players?

Solution :

8C5

Example 2 :

A committee of 5 people is to be chosen from a group of 6 men and 4 women. How many committees are possible if

there are no restrictions?
one particular person must be chosen on the committee?
one particular woman must be excluded from the committee?

Solution :

10C5
1.9C4
9C5

Example 3:

In a hand of poker, 5 cards are dealt from a regular pack of 52 cards.

In how many of these hands are there:

All hearts?

Solution : 13C5

All the same colour?

Solution: RED and BLACK

26C5 + 26C5 = 2.26C5

PERMUTATION & COMBINATION

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PROBABILITY

1. As a measure of chance

2. Single events (including listing all the possible outcomes in a simple chance situation to calculate the probability)

3. Simple combined events (including using possibility diagrams and tree diagrams, where appropriate)

4. Addition and multiplication of probabilities (mutually exclusive events and independent events)

Note and Practical Example :

Question 1 :

1) A die is rolled, find the probability that an even number is obtained?

Ans/Solution (Q.1)

· Let us first write the sample space S of the experiment.

S = {1,2,3,4,5,6}

· Let E be the event "an even number is obtained" and write it down.

E = {2,4,6}

We now use the formula of the Classical probability.

P(E) = n(E) / n(S) = 3 / 6 = 1 / 2

Question 2 :

2) There are 9 red marbles, 1 green marbles and 5 yellow marbles in a bag. Two marbles are drawn at a random from the bag,one after another without replacement. Some of the probability are shown below.

FIRST MARBLE

R = 9/15

G = 1/15

Y = 5/15

SECOND MARBLE

RR GR YR RR = 8/14 GR = 1/14 YR = 5/14

GR GG GY GR = 9/14 GG = 0 GY = 5/14

RY GY YY RY = 9/14 GY = 5/14 YY = 4/14

KEY:

R RED MARBLE

G GREEN MARBLE

Y YELLOW MARBLE

(a) Calculate both marble are red,

(b) One marble is red and the other is green

Ans/Solution (Q.2) :

(a) (9/15 * 8/14) = 12/35

(b) (9/15 * 1/14) + (1/15 * 9/14) = 6/35

= 59/105

PROBABILITY

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Measures of Central Tendency

The term "measures of central tendency" refers to finding the mean, median and mode.

Mean- Average

Median- Middle Value, when the data is arranged in numerical order,

Mode- The value (number) that appears the most.

Example 1:

Find the mean, median and mode for the following data:

5, 15, 10, 15, 5, 10, 10, 20, 25, 15.

(You will need to organize the data.)

5, 5, 10, 10, 10, 15, 15, 15, 20, 25

Example 2 :

On his first 5 biology tests, Bob received the following scores: 72, 86, 92, 63, and 77. What test score must Bob earn on his sixth test so that his average (mean score) for all six tests will be 80? Show how you arrived at your answer.

Possible solution:

Set up an equation to represent the situation. Remember to use all 6 test scores:

72 + 86 + 92 + 63 + 77 + x / 6 = 80

Cross multiply and solve: (80)(6) = 390 + x

480 = 390 + x

- 390 -390 / 90 = x

Bob must get a 90 on the sixth test.

MEASURE OF CENTRAL TENDENCY

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SETS

A collection of "things" (objects or numbers, etc). Each member is called an element of the set.

Example :

A group of people

{Tall,Fat,Short,Thin,Muscular}

Sets also can be grouped into a Venn diagram.

ԑ = {x : x is a positive integer <14}

A = {x : x divisible by 3}

B = {x : x divisible by 4}

Draw a Venn diagram to illustrate the information above.

ԑ = {1,2,3,4,5,6,7,8,9,10,11,12,13}

A = {3,6,9,12}

B = {4,8,12}

Set Notation :

∈ is an element of

∉ is not an element of

n(A) the number of elements in set A

n(A’) compliment of set A

∅ the empty set

ԑ universal set

⊆ is a subset of

⊂ is a proper subset of

∪ union

∩ intersection

Example 2 :

On the venn diagram below,shade the region representing the sets given.

(a) (A∪B)’

Ans/Solution :

SETS

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Introduction to Statistical Data.

The study of data: how to collect, summarize and present it.

How to collect data:

Descriptive (like "high" or "fast") or Numerical (numbers).

Numerical Data can be Discrete or Continuous:

Discrete data is counted,
Continuous data is measured(within a range)

Survey:

Step one: Create the questions
Step two: Ask the questions
Step three: Tally the results

Step four: present the results

How to Show Data

• Bar Graphs, the heights of the bars represent the frequency. The data is discrete

Vertical

Pie Charts, the angles formed by each part adds up to 360^o

Dot Plots, A graphical display of data using dots.

Line Graphs, a graph that shows information that is connected in some way (such as change over time)

Scatter (x,y) Plots,has points that show the relationship between two sets of data.

Pictographs

Histograms, it is a vertical bar graph with no gaps between the bars. The area of each bar is proportional to the frequency it represents.

Frequency Distribution, The organization of raw data in table form with classes and frequencies.

Stem and Leaf Plots, a diagram that summarises while maintaining the individual data point. The stem is a column of the unique elements of data after removing the last digit. The final digits (leaves) of each column are then placed in a row next to the appropriate column and sorted in numerical order.