![Mathematics 4 Combinatorics Part 1
PSHS Main Campus - Fourth Quarter 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!=362880
2!=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)= _
'(30-4)! 2bt --=(30X2eX28)(27)=657720
(26)(25) (3X2Xr)
To make the computation easier,
30t _ (30)(ze)(?8)(27)(26t)
: (30)(2e)(28 )(27) _ 657720
2tc! 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 YOUR
OWN IF YOU WAN1' ]'O END YOUR MATH 4 EXPERIENCE ON A POSI-]-IVE IJO1'E!
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M4 combinatronics hw
- 1. Mathematics 4 Combinatorics Part 1 PSHS Main Campus - Fourth Quarter Mr. Fortunato A."facuboy III Fundamerftcrl Countittg Prhciple or Multiplicntiou Prhtciple o.f Courttittg IfotreuettttlcctLrintndifferentrnt1s,attdmttlthcreoent(ittdepetden't"f.omthefirsteuutt)cttttoccru,in together 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 Plate Nunrbers Hozo ntmttl ordhLnnl Philipphrc plntc nwnltet's Ltre ttunilable? (at ordinnnl plnte numbet' cortsists of three letters follozued bt1 tlucc nunbers) E*p 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 are 26 choices {or 82 ancl 83. Take note that for at plate nr-rmber, the letters can be repeated. As for the digits, there are 10 digits (0-9). So for the fourth blank (B4), there are 10 choices; similarly there are 10 choiccs for 85 and 86. Jtrst like in the letters, the digits carn be repeated. Take note that in {iliirrg the blanks, it is seen as an application of the Multiplication Principle. Therefore, the answer rs (261(26)(26) (10)(10)(10) = 17 576 000. Ex2 Class Officers Corrsidcr ryour class roith 30 members. ln horu matlrl 'rurrys (N) cnn zoe selet't n prcsiderrt, uice'prasitTent, secr et ar ry sr ttl tr e nst.Lrcr ? E*p Usirrg the h'lank nrt'thod again. PVPS T Since there are 30 tnembers in the class aucl assuming there anyone is eligible to become an officer. So for the president - first blank (P), there are 30 choices. For the second blatrl< (vice-president), take note that a student cannot be elected to more than one position, therefore for VP there are 29 choices because the student elected as president cantrot be elected as vice-president also. Similarly, there are 28 choicesfol 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. AS Since there are 3 possibie airlines, for the first blarrk (A), there ale 3 choices. For the second blank (S), there are 4 choices. Take note now that to travel from Manila to Cebu, a person cannot take an airline and a sea vessel at the same time. Therefore, the Mr-rltiplication Principle CANNOT be used. This is an application of the Addition Principle. Page I of3
- 2. Mathematics 4 Combinatoricp Part 1 PSHS Main Campus - Fourth Quarter 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 and another 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 pairs Case l: l)hil-Thai pair, there are (5)(4) = 20 pairs -l: Case 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 Page 2 of 3
- 3. Mathematics 4 Combinatorics Part 1 PSHS Main Campus - Fourth Quarter 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!=362880 2!=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)= _ '(30-4)! 2bt --=(30X2eX28)(27)=657720 (26)(25) (3X2Xr) To make the computation easier, 30t _ (30)(ze)(?8)(27)(26t) : (30)(2e)(28 )(27) _ 657720 2tc! 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 YOUR OWN IF YOU WAN1' ]'O END YOUR MATH 4 EXPERIENCE ON A POSI-]-IVE IJO1'E! Page 3 of3