M4 combinatronics hw

Mathematics 4

PSHS Main Campus - Fourth Quarter

Combinatorics Part 1

Mr. Fortunato A."facuboy III

Fundamerftcrl Countittg Prhciple or Multiplicntiou Prhtciple o.f Courttittg

IfotreuettttlcctLrintndifferentrnt1s,attdmttlthcreoent(ittdepetden't"f.omthefirsteuutt)cttttoccru,intogether the truo euents cnn occur ht nut different roarls.

Strppose thnt there sre k euents. lf the first euent ma1 ocuu' in f\ r0a1/s, the second euent irt ft2 7.0t1Lls, the thirtl in ft., rotuls and

so ort t'ntd so .fortl't, then the le euents together nlau lcrrLt i/t ntn2n3 . . .nl_ ((;)ays.

Ex1

E*p

Plate Nunrbers Hozo ntmttl ordhLnnl Philipphrc plntc nwnltet's Ltre ttunilable? (at ordinnnl plnte numbet' cortsists ofthree letters follozued bt1 tlucc nunbers)

The solution is called the Blank Method. We use a blank to repr'esent each component of the plate ntimber.

81 82 83 84 85 86

The English alphabet consists of 26 letters. So for the fir,st blank (B1), there are 26 choices; similarly there are26 choices {or 82 ancl 83. Take note that for at plate nr-rmber, the letters can be repeated. As for the digits, thereare 10 digits (0-9). So for the fourth blank (B4), there are 10 choices; similarly there are 10 choiccs for 85 and86. Jtrst like in the letters, the digits carn be repeated. Take note that in {iliirrg the blanks, it is seen as anapplication of the Multiplication Principle. Therefore, the answer rs (261(26)(26) (10)(10)(10) = 17 576 000.

Class Officers Corrsidcr ryour class

secr et ar ry sr ttl tr e nst.Lrcr ?

Usirrg the h'lank nrt'thod again.

roith 30 members. ln horu matlrl 'rurrys (N) cnn zoe selet't n prcsiderrt, uice'prasitTent,Ex2

E*p

PVPS T

Since there are 30 tnembers in the class aucl assuming there anyone is eligible to become an officer. So for thepresident - first blank (P), there are 30 choices. For the second blatrl< (vice-president), take note that a studentcannot be elected to more than one position, therefore for VP there are 29 choices because the student electedas president cantrot be elected as vice-president also. Similarly, there are 28 choices fol S and 27 choices for T.This is another application of the Multiplication Principle. Therefore, the answer is (30)(29)(28)(27\ = 657 720.

Ex 3 Travel Optrons On n giuen tltu1, thert: are 3 differert nirlhrcs from Mnnila to Cclttt tuul 4 diffcrut sen uessels. Hozu

ttttnll/ waVS are there to go .front Mrmiltt to Cebu?

E*p Using tlre ['larrl< metlrod .rr:ain.

ASSince there are 3 possibie airlines, for the first blarrk (A), there ale 3 choices. For the second blank (S), there are4 choices. Take note now that to travel from Manila to Cebu, a person cannot take an airline and a sea vesselat the same time. Therefore, the Mr-rltiplication Principle CANNOT be used. This is an application of theAddition Principle.

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Mathematics 4

PSHS Main Campus - Fourth Quarter

Combinatoricp Part 1

Mr. Fortunato A. Tacuboy III

Adclitiorr Principle of Couttirtg

If one euent occur in m different roarls, and tutother auent can occw' irt n dit'ferent uays. If the truo eaents sccomplish the same

thing and cannot happen simultwreously \ruiually exclusiae eaents)') then the ttuo eaents corr occur in m + n dift'erent ruaqs.

Suppose thnt there ctre k mutunlly arclusizte eautts. If Lhe first eucnt ntnrl occttr itr n1 Lpfrys, the secontl euerft in rt, roays, the

third in n3 T.Ltatls srrd so on and so forth, then the ntLmber of zoays any o17e of the k eaents can occur is nt + n2 + n3 +' . .* flL)TPntls.

E*p Continr-ring with the explanation, for this problem, the solution is called the Case Method.

Case 1: Airline (3 possible choices)

Case 2: Sea Vessel (4 possible choices)

Since this is an application of the Addition Principle, the answer is 3 + 4 = 7 .

Ex 4 Plane Flights On s giuert clny, tlrcre rtre 5 clifferent clit'ect fliyqhts to Lord.on or tlrcrc nrc 2 JTi,gltts 7o Lft'nt! Kong ard 4

cornrcctirrg JTights J"rom Hong Kong to LondLm. Hrtrtt mantl rL)ar/s nrc there to go to London?

E*p Take note that the case method should be used.

Case l: direct flight ro London (5 possible choices)

Case 2: via Hong Kong

For Case 2, the Multiplication Principle should be used, there are two blanks: one for going to Hong Kong andanother from Hong Kor-rg to London. For Case 2, there are (2)(a) = 8 ways. So the answer is 5 + 8 = 13.

Ex 4 Subcommittee Au internntional committee corzsl>^f>^ of 5 representntiztes from the Philippittes, 6 from Malnysia nnd 4

from Thailnnd.

@) ht hout nTnrLy Llqys csn a subcommittee of three members be chosett, rro tzoo come from the sttme ctlunh'ry?

E*p Using the Multiplication Principle, the answer is (5)(6)(4) = 120.

'(B) In horo matty 70aL/s can n sultcommittee of tzuo members frctm different cotnttries?

' E"p Using the case method and using the blank rnethod for each case:

Case 1: Phil-Mal pair, there are (5)(6) = 30 pairsCase l: l)hil-Thai pair, there are (5)(4) = 20 pairsCase -l: Mal-Thai pair, there are (6)(4) = 24 pafts

Using the Addition Principle, the answer is 30 + 20 + 24 = 74.

HOME,WORK (due Monday)COLE p.775-777 #s 76,24,26, 38, 42

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Mathematics 4

PSHS Main Campus - Fourth Quarter

Combinatorics Part 1

Mr. Fortunato A. Taci-rboy III

Ex,rmple Recall:

Class Officers Consider yorLr class zoith 30 ntenrLtcrs. ln hozo tnafiV Taays (N) cnn tue sele ct n president, ttice-presidant, seuetarry

ntrd lrt'nsttt t't ?

To find the number of ways, it is possible that to first choose four people and etfter choosing these four people assign

their respective positions. This process is called a permutation.

Definition: Permutation - ordered arrangement, without repetition, of a set of objects

LetSbeasetof nelements and let 0<r <n. Thenumberof permutations of relements of Sis

,. P. = P(n, r) = t+ . ('The expression is read as "permtLtntiort rt takerr r" .){n-r}t

Take note that n! is read as n f:rctorial and is defined as: nl : n(n - 1)(n - 2) . . . (3X2)(1).

1!=1 3l=6 5!=120 7l=5040 9!=3628802!=2 4t=24 6!=720 8l=40320 10!=3628800

By definition, 0! : 1.

MEMORIZE THE FACTORTAI, VAL,UES OF THE FIRST 10 NATURAL NUMBERS!

E*p For the Class Officers ploblem, since it is an application of a permutation (n=30, r :4),

30! 30! (30)(29)"'{?)(?)r1\P(30,4)= _ --=(30X2eX28)(27)=657720'(30-4)! 2bt (26)(25) (3X2Xr)

To make the computation easier,

30t _ (30)(ze)(?8)(27)(26t) : (30)(2e)(28 )(27) _ 6577202tc! 26:

For exercise purposes, evaluate the following permutation expressions: P(5, 0), P(4, 1), P(9, 2\, P(6, 6), P(5, 4), (7 , 3) ,

P(10,7), P(13,5). Use your scientific calculators to vcrify your answers.

Take note of the following properties of P(n, r):

P(n, 0) = 1 P(n, n) = n/P(n, 1):ru P(n,n-7):n!P(n' 2) = rt2 - rr

HOMF.WORK (due Monday)VANCE p.282 #s 4, 6,8, p. 285 #s 6, B

Piece of Advice: Nofe thnt the nLrnilier of homeworh items is limited uthich menns that you do the

additionnl exercises 0n yutr lu)n. Ylu haue 3 textboolcs ruithloads of exercises and slme of the items

hsue ansrlers. YOU SHOULD HAVE'THE INITIATIVE'lO DO 1'HE EXERCISES ON YOUROWN IF YOU WAN1' ]'O END YOUR MATH 4 EXPERIENCE ON A POSI-]-IVE IJO1'E!

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