Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 M ARIO F. T RIOLA E IGHTH...

Copyright © 1998, Triola, Elementary StatisticsAddison Wesley Longman

1

MARIO F. TRIOLAMARIO F. TRIOLA EIGHTHEIGHTH

EDITIONEDITION

ELEMENTARY STATISTICS

Section 3-6 Counting


2

Assume a quiz consists of two questions. A true/false and a multiple choice with 5 possible answers. How many different ways can they occur together.


3

Tree Diagram of the events

a

b

c

d

e

a

b

c

d

e

T

F

T & aT & bT & cT & dT & eF & aF & bF & cF & dF & e

Assume a quiz consists of two questions. A true/false and a multiple choice with 5 possible answers. How many different ways can they occur together.


4

Tree Diagram of the events

a

b

c

d

e

a

b

c

d

e

T

F

T & aT & bT & cT & dT & eF & aF & bF & cF & dF & e

m = 2 n = 5 m*n = 10

Let m represent the number of ways the first event can occur.

Let n represent the number of ways the second event can occur.


5

Fundamental Counting Rule

For a sequence of two events in which the first event can occur m ways and the second event can occur n ways, together the events can occur a total of m • n ways.


6

An ATM pin number is a 4 digit number. How many possible pin numbers are there, if you allow repeats in each position?

Digit: 1st 2nd 3rd 4th

# of Choices: 10 10 10 10

By the FCR, the total number of possible outcomes are:

10 * 10 * 10 * 10 = 10,000

Example


7

An ATM pin number is made up of 4 digit number. How many possible outcomes are there, if no repeats are allowed?

Digit: 1st 2nd 3rd 4th

# of Choices: 10 9 8 7

By the FCR, the total number of possible outcomes are:

10 * 9 * 8 * 7 = 5040

Example


8

Rank three players (A, B, C). How many possible outcomes are there?

Ranking: First Second Third

Number of Choices: 3 2 1

By FCR, the total number of possible outcomes are:

3 * 2 * 1 = 6

( Notation: 3! = 3*2*1 )

Example


9

The factorial symbol ! denotes the product of decreasing positive whole numbers.

n! = n (n – 1) (n – 2) (n – 3) • • • • • (3) (2) (1)

Special Definition: 0! = 1

The ! key on your TI-8x calculator is found by pressing MATH and selecting PRB and selecting choice #4 !

Notation


10

An entire collection of n different items can be arranged in order n! different ways.

Example:

How many different seating charts could be made for a class of 13?

13! = 6,227,020,800

Factorial Rule


11

Eight players are in a competition, three of them will win prizes (gold/silver/bronze). How many possible outcomes are there?

Prizes: gold silver bronze

Number of Choices: 8 7 6

By FCR, the total number of possible outcomes are:

8 * 7 * 6 = 336 = 8! / 5!

= 8!/(8-3)!

Example


12

n is the number of available items (none identical to each other)

r is the number of items to be selected

the number of permutations (or sequences) is

Permutations Rule

Order is taken into account

Pn r = (n – r)!n!


13

when some items are identical to othersIf there are n items with n1 alike, n2 alike, . . .

nk alike, the number of permutations is

Permutations Rule

n1! . n2! .. . . . . . . nk! n!


14

How many ways can the letters in MISSISSIPPI be arranged?

I occurs 4 times

S occurs 4 times

P occurs 2 times

Permutations Rule

4!4!2! 11!

=34,650


15

Eight players are in a competition, top three will be selected for the next round (order does not matter). How many possible choices are there?

By the Permutations Rule, the number of choices of Top 3 with order are 8!/(8-3)! = 336

For each chosen top three, if we rank/order them, there are 3! possibilities.

==> the number of choices of Top 3 without order

are {8!/(8-3)!}/(3!) = 56

Example

8!

(8-3)! 3!=

Combinations rule


16

n different items

r items to be selected

different orders of the same items are not counted

the number of combinations is

(n – r )! r!n!

nCr =

Combinations Rule


17

Permutation –Order Matters

Combination - Order does not matter

TI-83/4

Press MATH choose PRB choose 2: nPr or 3: nCr to compute the # of outcomes.

Example:

10P5 = 10 nPr 5 = 30240

10C5 = 10 nCr 5 = 252


18

Is there a sequence of events in which the first can occur m ways, the second can

occur n ways, and so on?

If so use the fundamental counting rule and multiply m, n, and so on.

Counting Devices Summary


19

Are there n different items with all of them to be used in different arrangements?

If so, use the factorial rule and find n!.



20

Are there n different items with some of them to be used in different arrangements?

If so, evaluate


(n – r )!n!

n Pr =


21

Are there n items with some of them identical to each other, and there is a need to find the total number of

different arrangements of all of those n items?

If so, use the following expression, in which n1 of the

items are alike, n2 are alike and so on


n!n1! n2!. . . . . . nk!


22

Are there n different items with some of them to be selected, and is there a need to find the total number of combinations (that is, is the order irrelevant)?

If so, evaluate


n!nCr = (n – r )! r!

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 M ARIO F. T RIOLA E IGHTH...

Documents

Transcript of Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 1 M ARIO F. T RIOLA E IGHTH...