1. HHl"y,9,v,,*c"#i*.
fm Use the fundamental eounting principle and permutations.
Vocabulary
An ordering
ofz objects a permutationofthe objects.
is
The symbol ! is the factorial slrrnboldefinedasfollows:
nr = 2. (n - l). (, - 2)..... 3 . 2 . l
f$ffinm Use the fundamental counting principle
Dining.A restaurant
offers5 appetizers,4
salads, entrees, 7 desserts.
11 and How
manydifferentwayscanyou ordera completemeal?
Solution
Youcaausethe firndamentalcountingprinciple to find the total numberofways to
ordera completemeal.Multiply the numberof appetizers the number
(5), of salads
(4), thenumber entrees
of (11),andthenumber oia".r"rts 17;.
Nu mber ofways:5 . 4. 11.7 :1 5 4 0
Thenumber different
of waysyou canordera complete
mealis 1540.
ffirffi Use the counting principle with repetition
Fundraising A raffle ticket contains1 digit followed by 3 letters.
(a) How many
arepossible letters digits berepeateai nowmany
if and can
different :lFl' arepossibleif lettersand digits carurotU" ."peut")-i
9lP::11tickets 1u;
Solution
a. Therere 26 choices each for letterand 10choices eachdigit.
for
. You canusethe fundamental
countingprinciple to find the number
of differenttickets.
Number oftickets: l0 . 26 . 26 .26 : 175,760
With repetition, number different
the of ticketsis 175,760.
b. If you cannotrepeatletters,thereare26 choicesfor the first letter,
but then only 25 choicesfor the secorld letter,and24 choicesfbr
the third letter Therearestill l0 choicesfor the digit. you canuse
the fundamental countingprinciple to find the nuniberof
different tickets.
Number oftickets: l0 . 26 . ZS. 24: 156,000
Withoutrepetition, number different
the of ticketsis 156,000.
Exercises for Examples 1 and 2
1. A deli sells4 sizes ice cream
of cones (small,medium, large, giant)and
and
2 different cones(waffle and cake).How manychoicesdoJsthe
Oetioffert
2,
{ 1aIfleticket conlains4 digits followed by 2 leters. (a) How manydifferent
tickets arepossibleif lettersand digits canbe repeatedi(b) How
manydifferent
ticketsarepossible letters digitscannot repeatedi
if and be
Algebra
2
. Chapter Hesource
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Find the likelihood that an event will occur.
Vocabulary.
The probability of an eventis a numberfrom 0 to 1 that indicatesthe
likelihood event
the will occur.
When all outcomes equallylikely, the theoretical probability that
are
an eventwill occuris
Numberofoutcomesin event
I
r('4 ) : --Totuftffier of ouGomes '
Odds measure likelihood that an eventwill occur.
the
The experimental probability of an eventI is givenby
: Numberoftrials where occurs
I
f(li - '
Totul nGbet o-fttiul,
A geometricprobability is a ratio of two lengths,areas, volumes.
or
m@ Find probabilities of events
There ar€ eight balls in a bag. Three are red, three are yellow, and two
a?6green. Find the probability of choosing a gteen ball.
Thereare 8 possibleoutcomes.
Number of ways to choose a green ball 2l
I(choosinga greenball) : Numberof outcomes 84
Etrtr[E&:lUse permutationsor combinations
Auditions For a play tryout, 10 students reciting monologuesThe orderin which
are
the studentsperfom is randomlyselected. What is the probability that the students
(a)
auditionin alphabetical orderby last name?(b) What is the probability that 2 of your
4 friendstrying otlt will be the fust 2 performers?
a, Thereare10! differentp of Only l is
ermutations the 10 studenls.
orderby last name.
in alphabetical
ll
order): l-or : t6r&800 - 0.000000276
P(alphabetical
, b. Thereare ,oC, different combinationsoftwo students. these
Of
oC, are2 ofyour friends.
,C" A )
P(fust2 students your friendsl: =
are
Fr:; 15- 0.133
Exercises for Examples 1 and 2
1. Find the probability of choosinggnered ball in Example I .
2. Find the probability that the fust 3 performersareyour ftiends in Example2.
2
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L_j!:!_l Forusewith
mflntr Findthe numberof permutations
Track Meet Eight runnerscompete an elite track meetcompetition.(a) Assuming
in
thereareio ties, in how manywayscould the runnersfinish in the meet?(b) Assuming
thereareno ties, in how manywayscould the runnersplacefirst, secon4or third in
themeet?
Solution
a. Thereare8! differentwaysthat the runnerscanfinish.
8! : 8. 7 . 6. s. 4 . 3 . 2 . 1 : 4 0 , 3 2 0
b. Any of the 8 rumers canfinish fust, then any of the remaining
7 runnerscanfinish second, any ofthe remaining6 runners
and
canfinish third. So the numberof waysthat the runnerscanplace
second thirdis 8.7 .6:336.
first. or
ruEru Find permutations of .n obiects taken r at a time
Baseball A baseballteamhas 14players.Hbw many9-playerbatting orderscan
be formed?
Solution
Find the numberofpermutationsof 14playerstaken9 at a time.
tnn
l4'! = ]4 - 87,178.',ot :
P - 726,485'760
e r 14 - 9)!
14: st t;::1
The team can form 726,485,760 batting orders with 9 players.
HU!ffi Find permutataonswith repetition
Find the number of distinguishable permutations of the letters in
(al DELAWAREand (bl PHILADELPHIA.
a. DELAW.ARE 8 lettersofwhich A and E areeachrepeated
has
2 times.So,thenumber distinguishable
of permutations
is
R' 40.320
2,.at= q : 10'080'
b. PHILADELPHIAhas12letters whichB H, I, L, andA are
of
eachrepeated times. So,the numberof distinguishable
2
. t2! 479'qo
'600: 14.968,800.
ls
Permurallons -
t! .2r. . 2r.' 2r . 2!
Exercises for Examples 3,4, and 5
3. ReworkExample3 if 9 runnerscompete.
4. ReworkExample if thebaseball
4 teamhasl2 players.
Find the number of distinguishable permutations of the !€tters in
the word.
5. ALABAMA 6. STREET 7. MISSOURI
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Exercises for Examples 1 and 2
Find the number of combinations.
1, uCo 2. ,oCt 3. oC" 4' ,tcr.
5. ReworkExample2 to find how manywaysyou cango to at leasttwo of the
four football eames.
Use Pascal's triangle
CommitteeMembers Usepascalttriangle find thenumber combinations
to of of
3 committeememberschosenfrom g availablemembers.
Solution
To find rC, write the 8th row of pascal,s
triangle.
n:7 (Tthrow) 1 721 353521 71
z:8 (8throw) I 8 28 s67056288 I
'
rco ,c, ,c, ,c, ,Co ,c, ,cu ,c, ,C,
ThevalueofrC, is the4th number the gthrow ofpascal,s
in triangle,
so = 56.
,C,
Thereare 56 combinations 3 committeemembers.
of
mGElfi Expanda power of a binomial difference
Use the binomial theorem to write the binomial expansion.
tz - ?t l = I u + r - 1113
J_,I
t-
:
3c0t(-3)0 + 3ct*{ rt + 3c2zt(-3)2 3q{.r3
+
= (rxrxi) + (r(?)C, + (3XzXe)
+ (1XlX_27)
:23-9?+272- 2 7
trWr'ffii Find a coefficient in an expansion
Find the coefficient of x3 in the expanaion of (4x + 315.
Eachtermin theexpansion the form
has - ,e)r.The termcontaining:r3
sc!4is
occurswhenr:2:
=
sc2(4i3(3)2 0o)(64x3)( = 576ox3
Thecoefficient -r3is 5760.
of
Exercisesfor Examples 3,4, and S
6. ReworkExample3 choosing4 committeemembers.
7. Usethebinomialtheorem expand expression + Z)a.
to the (x3
8. Findthe coeffrcient the.x2_term Example .
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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 fives | .-
Number ofnon-fives i, or r:)
b. Odds
against
gelling 5
a :
f. or 5:t
Ifn:fililtfi! Find an experimental probability
The table shows the results of tossing two
coins twenty times. Find the experimental
probability of getting a head on the first toss. HH 3
Theexperimental p
probability ofgettinga headon HT 6
thefust tossis the sumof HH andHT.
?+ 5 0 TH 5
fl nead on lst toss) -- - -20" = : 0.45
20 TT 6
,ly.Fr,nf*t Find a geometric probability
You throw a beanbag at a square board shown. F_4ft __ 1
Your beanbag is equally likely to hit any point on
the board. ls the bag more likely to land outside T
the smaller square, or inside the smaller square?
Areaoutside
smailsquare JN
P(landing outsidesmall square)-
Areaofentireboard
_42 -32 _'I I
I
42 16
P(landing
inside
small = A':i++-:d]j!"=
square) : { : :
Areaol entireboard 42 l6
because <
-7 9
t6 16.you aremorelikely to landinsidethesmaller
square.
Exercisesfor Examples 3,4, and 5
A card is drawn from a standard deck. Find the odds.
3. Againstdrawing king
a 4. In favorof drawing club
a
5. Findtheexperimental
probability gettinga tail on the second in
of toss
Example4.
6, ReworkExample for squares
5 with sides 2 feetand3 feet.
of
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