Counting Techniques

REVIEWPERMUTATION/COMBINATION

Counting Techniques

#1 How many times are you asked to choose/decide?

Counting Techniques

#2For each decision you make, how many choices available?

Sample Space

A sample space is made up of all possible outcomes in an experiment.

Fundamental Principle of Counting (FCP)

If one thing can occur in m ways and a second thing in n ways, and a third thing in p ways, and so on, then the sequence of things can occur in (m)(n)(p)... ways.

Example 1Three coins are tossed. How many outcomes are possible?

8 outcomes

Example 2

A movie theater sells 4 sizes of popcorn (S, M, L, and XL) with 3 choices of flavors (BBQ, Cheese and Butter). How many possible ways can a bag of popcorn be purchased? 12 ways

Example 3

A car number plate contains three letters followed by three digits.

17,576,000 car number plates

How many car number plates can be made if letters and digits can be repeated?

Example 4

A car number plate contains three letters followed by three digits.

11, 232,000 car number plates

How many car number plates can be made if letters and digits cannot be repeated?

Example 5

Suppose a 3-digit number is to be formed from the digits 1, 2, 4, 5, 7, 8 and 9. How many numbers can be formed if

343 numbersrepetition is allowed?

Example 6

Suppose a 3-digit number is to be formed from the digits 1, 2, 4, 5, 7, 8 and 9. How many numbers can be formed if

210 numbersrepetition is not allowed?

Example 7

Suppose a 3-digit number is to be formed from the digits 1, 2, 4, 5, 7, 8 and 9. How many numbers can be formed if

147 numbersthe numbers to be formed are even?

Example 8

Suppose a 3-digit number is to be formed from the digits 1, 2, 4, 5, 7, 8 and 9. How many numbers can be formed if

196 numbersthe numbers to be formed are odd?

Example 9

How many 3-digit numbers greater than 700 can be formed using the digits 5, 6, 7, 8 and 9 if

75 numbersrepetition is allowed?

Example 10

How many 3-digit numbers greater than 700 can be formed using the digits 5, 6, 7, 8 and 9 if

36 numbersrepetition is not allowed?

Example 11

How many 3-digit numbers greater than 700 can be formed using the digits 5, 6, 7, 8 and 9 if

30 numbersthe numbers must be even?

Example 12

How many 3-digit numbers greater than 700 can be formed using the digits 5, 6, 7, 8 and 9 if

45 numbersthe numbers must be odd?

Factorial Notation

The factorial notation (!) is a notation usually represented by n! (read as n-factorial) and defined as

n! = n (n-1) (n-2) (n – 3) …. (3) (2) (1)

Factorial Notation

PRACTICE

5! = 5(4)(3)(2)(1)120

Factorial Notation

PRACTICE

8! = 8(7)(6)(5)(4)(3)(2)(1)40, 320

Factorial Notation

PRACTICE

20! = 2.43x1018

Factorial Notation

PRACTICE

6! 4!

= 6(5)(4)(3)(2)(1) 4(3)(2)(1) = 30

Factorial Notation

PRACTICE

15! 5!

= 1.09x1010

Factorial Notation

PRACTICE

0! = 1

Example 13

In how many ways can 7 persons be lined up to get on a bus?

5040 ways

Example 14

How many different arrangements can we form from the word MAPHY?

120 arrangements

Example 15

In how many ways can 5 persons lined to get on a bus if a certain

48 ways2 persons must follow each other?

Example 16

In how many ways can 5 persons lined to get on a bus if a certain

36 ways3 persons must follow each other?

Example 17

In how many ways can 5 persons lined to get on a bus if a certain3 persons must not follow each other?

COMPLEMENTARY EVENTS

Complementary Events

All outcomes that are not the event.

Complementary Events

When the event is heads, the complement is tails.

When the event is Mon and Fri, the complements are Tue, Wed, Thurs, Sat and Sun

When the event is spades, the complements are hearts, club and diamonds

Example 17

In how many ways can 5 persons lined to get on a bus if a certain3 persons must not follow each other?5! – (3!)(3!)The likelihoood

that the event will happen120 – 36 = 84 ways

Example 18

Four married couples have bought 8 seats in a row for a concert. In how many ways can they be seated

40320 wayswithout any restrictions?

Example 19

Four married couples have bought 8 seats in a row for a concert. In how many ways can they be seated

384 waysif each couple is to sit together?

Example 20

Four married couples have bought 8 seats in a row for a concert. In how many ways can they be seated

1152 waysif all men and women sit together?

Example 21

Four married couples have bought 8 seats in a row for a concert. In how many ways can they be seated

576 waysif all men sit together to the right of women

Example 22

In how many ways can 4 Americans, 5 Chinese, and 6 Filipinos be seated

1.31x1012 waysin a row?

Example 22

In how many ways can 4 Americans, 5 Chinese, and 6 Filipinos be seated

12, 441, 600 waysif each nationality sit together?

Assignment

1.) Bring your textbook tomorrow.2.) Reporting by group

- page 455 – 456 except numbers 11 -12

- 6 groups- Visuals are graded- UNDERSTANDING

CriteriaVisuals - 5 pointsExplanation - 10 pointsAbility to answerQuestions - 5 pointsPreparednessand Poise - 5 points

Total - 25 points

Review 1

How many possible outcomes result in rolling a fair dice three times?

216 outcomes

Review 2

How many ways can you choose a 3-digit pin number where repetition is allowed?

1000 ways

Review 3

How many ways can you choose a car plate number composed of 3 letters excluding vowels and 3 digits if repetition of letters and digits are not allowed?5,745,600 ways

Review 4

How many times can a cellular network come up with different cellphone numbers composed of 11 digits if the first 4 digits must be 0923 and the number must be even?5,000,000 ways

RETURN

Review 53 males and 3 females lined up to get on a bus. How many ways can they be arranged

720 ways

without restriction?

Review 63 males and 3 females lined up to get on a bus. How many ways can they be arranged

48 ways

if a male and a female sit together ?

Review 73 males and 3 females lined up to get on a bus. How many ways can they be arranged

36 ways

if all women sit together to the right of men?

Review 8

How many ways can the students be lined up in a school production number composed of 2 freshmen, 2 sophies 3 juniors and 4 seniors,

39, 916, 800 wayswithout any restriction?

Review 9How many ways can the students be lined up in a school production number composed of 2 freshmen, 2 sophies 3 juniors and 4 seniors,

13,824 ways

if students of the same year level have to stand next to each other?

Review 10How many ways can the students be lined up in a school production number composed of 2 freshmen, 2 sophies 3 juniors and 4 seniors,

39,902,976 ways

if students of the same year level MUST NOT stand next to each other? RETURN

Permutation and Combination

PermutationHow many 4-digit pin numbers can you make if repetition is not allowed?

CombinationHow many different handshakes are possible in a room with 4 people?

5040 pin numbers

6 handshakes

Permutation and Combination

Permutationis an arrangement following a definite order. The way things are arranged is considered.Combinationis a selection made from a group of items without regard to their order.

Permutation )!rn(

!nr,nPPnr

)!rn(!nr,nPPn

r )!rn(

!nr,nPPnr

nPr = P

nr = P(n,r) =

n!(n-r)!Combination

n!r!(n-r)!

nCr = C

nr = C(n,r) = ( ) n

r

=

Example 1:

The ski club with ten members is to choose three officers, namely: captain, co-captain and secretary. How many ways can those offices be filled?720 ways

Permutation

Example 2:

You are on your way to an island for a vacation and of 8 possible books, your parents say you can only take 3. How many different collection of 3 books can you take?56 collections

Combination

Example 3:

A school contest requires each section of 20 students to select 5 representatives. How many ways can this be done?

15,504 ways

Combination

Example 4:

There are 12 standbys who hope to get on the flight to Camiguin, but only 6 seats are available. How many different ways can the 6 people be selected?924 ways

Combination

Example 5:

In the production of Snow White and the Seven Dwarfs, 15 actors are considered for the roles of the seven dwarfs. How many ways can the director cast the roles?32, 432, 400 ways

Permutation

Example 6:

Suppose you are asked to list, in order or preference, the three best movies you have seen this year. If you saw 10 movies during the year, in how many ways can the three best be ranked?720 ways

Permutation

Example 7:

During the HS Intramurals, the volleyball tournament requires each year level to play every other year level once. How many matches are possible?

6 matches

Combination

Permutation

In how many ways can 4 boys and 3 girls be seated in a row of 5 chairs?

2,520 ways

Permutation

In how many ways can six students participating in the math quiz bowl be ranked 1st, 2nd and 3rd place?

120 ways

Permutation

In how many ways can a coach assign the starting positions in a basketball game to nine equally qualified men?

60, 480 ways

Permutation

A company advertises two job opening, one for an announcer and the other for a new writer. If 8 people who are qualified for either position applied for the job, in how many ways can the opening be filled?

56 ways

Permutation of Objects which are

not distinctHow many different arrangements can you make from the word POOP?

Permutation of Objects which are

not distinctThe permutation of n objects where n1 are of one kind, n2 ok a second kind, n3 of the third, …, and nk of a kth kind isn!

n1!n2!n3!...nk!

Permutation of Objects which are not distinct

ALGEBRA

2, 520 ways

7! 2!1!1!1!1!1!

7! 2!

=

Permutation of Objects which are not distinct

STATISTICS

50, 400 ways

10! 3!3!2!

Permutation of Objects which are not distinct

INFINITY

50, 400 ways

10! 3!3!2!

Permutation

How many signals can be made by arranging 8 flags in a line, 4 are red, 2 are blue and 2 are white?

420 ways

Permutation

Given 4 bulbs, three are red and one green. Determine the number of ways of arranging the bulbs in a string of wire.

4 ways

Permutation

In how many ways can 4 persons (A, B, C and D), arrange themselves in a circular table?

6 ways

Permutation

In how many ways can 7 different seedlings be planted in a circle?

720 ways

Permutation

Ten people are to sit at a round table. In how many ways can they be seated without restrictions?

362, 880 ways

Permutation

Ten people are to sit at a round table. In how many ways can they be seated if the host and hostess are to sit next to each other?

80, 640 ways

Permutation

In how many ways can 6 ladies be seated at a round table without any restrictions?

120 ways

Permutation

In how many ways can 6 ladies be seated at a round table if 4 particular ladies must sit next to each other?

48 ways

Permutation

In how many ways can 6 ladies be seated at a round table if 3 particular ladies must NOT sit next to each other?

84 ways

Combination

A committee of 5 members will be selected from a class of 15 students. How many committees are possible?

3003 committees

Combination

In a deck of cards, how many ways can we select 5 diamonds?

1287 ways

Combination

In a lotto with 49 numbers, how many ways can we select 6 numbers?

13, 983, 816 ways

Combination

A shipment of 12 television sets contain 4 defective sets. In how many ways can a hotel purchase five of these sets and receive

336 ways2 defective sets?

Combination

A shipment of 12 television sets contain 4 defective sets. In how many ways can a hotel purchase five of these sets and receive

456 waysat least 2 defective sets?

Combination

How many groups of less than four members can be formed from 10 boys?

165 ways

Combination

A box contains 5 red, 4 blue, and 3 white balls. In how many ways can we select 3 balls such that

60 waysthey are of different colors?

Combination

A box contains 5 red, 4 blue, and 3 white balls. In how many ways can we select 3 balls such that

18 waystwo are blue and 1 white?

Combination

A box contains 5 red, 4 blue, and 3 white balls. In how many ways can we select 3 balls such that

48 waysexactly 2 are blue?

Combination

A box contains 5 red, 4 blue, and 3 white balls. In how many ways can we select 3 balls such that

56 waysnone is blue?

Assignment1.) Bring your textbook on Monday

2.) On your lecture notebook, answer

- #’s 33 – 38 page 465

(no need to make a tree diagram)

- #’s 9 – 16 pages 470 - 471