HHl"y,9,v,,*c"#i*.fm Use the fundamental eounting principle and permutations.

VocabularyAn ordering ofz objects is a permutation ofthe objects.The symbol ! is the factorial slrrnbol defined as follows:nr = 2. (n - l ) . ( , - 2) . . . . . 3. 2. l

f$ffinm Use the fundamental counting principleDining.A restaurant offers 5 appetizers,4 salads, 11 entrees, and 7 desserts. Howmany different ways can you order a complete meal?SolutionYou caa use the firndamental counting principle to find the total number ofways toorder a complete meal. Multiply the number of appetizers (5), the number of salads(4), the number of entrees (11), and the number oia".r"rts 17;.Number ofways :5 . 4. 11 .7 :1540

The number of different ways you can order a complete meal is 1540.

Use the counting principle with repetitionffirffiFundraising A raffle ticket contains 1 digit followed by 3 letters. (a) How many9lP::11 :lFl'

are possible if letters and digits can be repeateai 1u; now manydifferent tickets are possible if letters and digits carurot U" ."peut")-iSolution

a. There re 26 choices for each letter and 10 choices for each digit.. You can use the fundamental counting principle to find the numberof different tickets.Number oftickets : l0 . 26 . 26 .26 : 175,760With repetition, the number of different tickets is 175,760.

b. If you cannot repeat letters, there are 26 choices for the first letter,but then only 25 choices for the secorld letter, and 24 choices fbrthe third letter There are still l0 choices for the digit. you can usethe fundamental counting principle to find the nuniber ofdifferent tickets.Number oftickets : l0 . 26 . ZS . 24: 156,000Without repetition, the number of different tickets is 156,000.

Exercises for Examples 1 and 21. A deli sells 4 sizes of ice cream cones (small, medium, large, and giant) and

2 different cones (waffle and cake). How many choices doJs the Oeti offert2,

{ 1aIfle ticket conlains 4 digits followed by 2 leters. (a) How many differenttickets are possible if letters and digits can be repeatedi (b) How many differenttickets are possible if letters and digits cannot be repeatedi

Algebra 2. Chapter 10 Hesource Book

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@f,,.,y#t,9",H-',*"Find the likelihood that an event will occur.

Vocabulary.The probability of an event is a number from 0 to 1 that indicates thelikelihood the event will occur.

When all outcomes are equally likely, the theoretical probability thatan event will occur is

Number ofoutcomes in event Ir('4 ) : --Totuftffier

of ouGomes '

Odds measure the likelihood that an event will occur.

The experimental probability of an event I is given by

Number oftrials where I occursf(li : -

Totul nGbet o-f ttiul, '

A geometric probability is a ratio of two lengths, areas, or volumes.

m@ Find probabilities of events

There ar€ eight balls in a bag. Three are red, three are yellow, and twoa?6 green. Find the probability of choosing a gteen ball.

There are 8 possible outcomes.Number of ways to choose a green ball

I(choosing a green ball) : Number of outcomes2l84

Etrtr[E&:l Use permutations or combinationsAuditions For a play tryout, 10 students are reciting monologues The order in whichthe students perfom is randomly selected. (a) What is the probability that the studentsaudition in alphabetical order by last name? (b) What is the probability that 2 of your4 friends trying otlt will be the fust 2 performers?

a, There are 10! differentp ermutations of the 10 studenls. Only l isin alphabetical order by last name.

l lP(alphabetical order) : l-or

: t6r&800 - 0.000000276

, b. There are ,oC, different combinations oftwo students. Of these

oC, are 2 ofyour friends. ,C" A )

P(fust 2 students are your friendsl : Fr:;

= 15 - 0.133

Exercises for Examples 1 and 2

Find the probability of choosing gne red ball in Example I .

Find the probability that the fust 3 performers are your ftiends in Example 2.

1.

2.

Algabra 2Chapter 10 Resource Book ' 3t

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mflntr Find the number of permutations

Date

Track Meet Eight runners compete in an elite track meet competition. (a) Assumingthere are io ties, in how many ways could the runners finish in the meet? (b) Assumingthere are no ties, in how many ways could the runners place first, secon4 or third inthe meet?

Solution

a. There are 8! different ways that the runners can finish.8! : 8. 7 . 6. s. 4. 3. 2. 1 : 40,320

b. Any of the 8 rumers can finish fust, then any of the remaining7 runners can finish second, and any ofthe remaining 6 runnerscan finish third. So the number of ways that the runners can placef i rst . second or third is 8.7 .6:336.

ruEru Find permutations of .n obiects taken r at a timeBaseball A baseball team has 14 players. Hbw many 9-player batting orders canbe formed?

SolutionFind the number ofpermutations of 14 players taken 9 at a time.

P - l4'! = ]4 - 87,178.',ot tnn14: e r 14 - 9)! st t;::1

: 726,485'760

The team can form 726,485,760 batting orders with 9 players.

HU!ffi Find permutataons with repetition

Find the number of distinguishable permutations of the letters in(al DELAWARE and (bl PHILADELPHIA.

a. DELAW.ARE has 8 letters ofwhich A and E are each repeated2 times. So, the number of distinguishable permutations is

R' 40.3202,. at= q : 10'080'

b. PHILADELPHIA has 12 letters of which B H, I, L, andA areeach repeated 2 times. So, the number of distinguishable

. t2!Permurallons ls t! .2r. . 2r. ' 2r . 2!

- 479'qo '600 : 14.968,800.

Exercises for Examples 3,4, and 53. Rework Example 3 if 9 runners compete.

4. Rework Example 4 if the baseball team has l2 players.

Find the number of distinguishable permutations of the !€tters inthe word.

5. ALABAMA

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f .n.l-Study GuidG) continued| .v.- | For use with pages 690_$97

Exercises for Examples 1 and 2

Find the number of combinations.

1, uCo 2. ,oCt 3. oC" 4' ,tcr.

5. Rework Example 2 to find how many ways you can go to at least two of thefour football eames.

Use Pascal's triangleCommittee Members Use pascalt triangle to find the number of combinations of3 committee members chosen from g available members.Solution

To find rC, write the 8th row of pascal,s triangle.

n:7 (Tthrow) 1 721 353521 71z:8 (8th row) I 8 28 s67056288

rco ,c, ,c, ,c, ,Co ,c, ,cu ,c,' The value ofrC, is the 4th number in the gth row ofpascal,s triangle, so ,C,There are 56 combinations of 3 committee members.

mGElfi Expand a power of a binomial differenceUse the binomial theorem to write the binomial expansion.tz - ?t l = Iu + r-1113

t - \ J_, I

: 3c0t(-3)0 + 3ct*{ rt + 3c2zt(-3)2 + 3q{.r3= (rxrxi) + (r(?)C, + (3XzXe) + (1XlX_27):23-9?+272-27

I

,C,= 56.

trWr'ffii Find a coefficient in an expansion

Find the coefficient of x3 in the expanaion of (4x + 315.Each term in the expansion has the form sc!4is

- ,e)r.The term containing:r3occurswhenr:2:

sc2(4i3(3)2 = 0o)(64x3)(\ = 57 6ox3

The coefficient of -r3 is 5760.

Exercises for Examples 3,4, and S6. Rework Example 3 choosing 4 committee members.

7. Use the binomial theorem to expand the expression (x3 + Z)a.

8. Find the coeffrcient of the .x 2_term in Example 5 .Algebra 2Chapter 10 Resource Book2

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milqr+ Find odds

A six-sided die'is tossed. Find (al the odds in favor of getting a 5 and(bl the odds agarnst getting a 5.

a. Odds in favor ofgetting a 5 : Number of fivesNumber ofnon-fives

| . -i, or r:)

b. Odds against gel l ing a 5 : f . or 5: t

Ifn:fililtfi! Find an experimental probability

The table shows the results of tossing twocoins twenty times. Find the experimentalprobability of getting a head on the first toss.The experimental probability p ofgetting a head onthe fust toss is the sum of HH and HT.

?+5 0fl nead on lst toss) -- - -20" =

20 : 0.45

,ly.Fr,nf*t Find a geometric probability

HH 3

HT 6

TH 5

TT 6

You throw a beanbag at a square board shown.Your beanbag is equally likely to hit any point onthe board. ls the bag more likely to land outsidethe smaller square, or inside the smaller square?

P(landing outside small square) - Area outside smail square

Area ofentire board

_42 -32 _ ' I42 16

F_4ft __ 1

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P(landing inside small square) = A':i++-:d]j!"= : { : :Area ol entire board 42 l6

-79because t6 <

16. you are more likely to land inside the smaller square.

Exercises for Examples 3,4, and 5

A card is drawn from a standard deck. Find the odds.3. Against drawing a king 4. In favor of drawing a club5. Find the experimental probability of getting a tail on the second toss in

Example 4.

6, Rework Example 5 for squares with sides of 2 feet and 3 feet.

Algebra 2Chapter 10 Besource Book32