Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology

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1 Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a floc of 6 pigeos flies ito a set of 5 pigeoholes to oost What ca we coclude? 3 Ageda Ch 3. The Pigeohole Piciple Pigeohole Piciple Geealized Pigeohole Piciple Applicatios Ch 3.4 Biomial Coefficiets Biomial Theoem Pascal's Idetit y ad Tiagle Theoem elated to Biomial Coefficiets Pigeohole Piciple A least oe of these 5 pigeoholes must have at least two pigeos i it Because thee ae 6 pigeos but oly 5 pigeoholes This is Pigeohole Piciple 4

2 Pigeohole Piciple Pigeohole Piciple If is a positive itege ad + o moe obects ae placed ito boxes, the thee is at least oe box cotaiig two o moe of the obects Also c alled the Diichlet Dawe Piciple the ieteeth-cetuy Gema mathematicia Diichlet Poof by cotapositio (p q q p) Suppose that oe of the boxes cotais moe tha oe obect The the total umbe of obects would be at most This is a cotadictio Pigeohole Piciple Example How may wods we should have if thee must be at least two that begi with the same lette? 7 Eglish wods, because 6 lettes i the Eglish alphabet Example How may people we should have if thee must be at least two with the same bithday? 367 people because 366 possible bithdays 7 Pigeohole Piciple Coollay A fuctio f fom a set with + o moe elemets to a set with elemets is ot oeto-oe a b c d 3 6 Geealized Pigeohole Piciple Pigeohole Piciple states that if + o moe obects ae placed ito boxes, the thee is at least oe box cotaiig two o moe of the obects How about if we have + obects? 3 + obect? + obect? 8 5

3 Geealized Pigeohole Piciple Geealized Pigeohole Piciple If N obects ae placed ito boxes, the thee is at least oe box cotaiig at least N/ obects Poof by Cotadictio Suppose that oe of the boxes cotais moe tha N/ - obects The total umbe of obects is at most N N < + N N/ < (N/) + This is a cotadictio because thee ae a total of N obects Geealized Pigeohole Piciple Accodig to geealized pigeohole piciple, whe we have N obects, thee must be at least obects i oe of the boxes as log as N/ N, whee N ( - ) +, is the smallest itege satisfyig N/ N? Geealized Pigeohole Piciple A commo type of poblem ass fo the miimum umbe of obects such that at least of these obects must be i oe of boxes whe these obects ae distibuted amog the boxes? 0 Geealized Pigeohole Piciple N/, N ( - ) +, is the smallest itege satisfyig N/ Could a smalle value of N suffice? No If ( - ) obects We could put - of them i each of the boxes No box would have at least obects 9

4 Geealized Pigeohole Piciple Example N/ N ( - ) + How may people out of 00 people wee bo i the same moth? N 00? 00/ 9 who wee bo i the same moth 3 Geealized Pigeohole Piciple Example 3 N/ N ( - ) + Show that amog ay + positive iteges ot exceedig thee must be a itege that divides oe of the othe iteges Assume we have + iteges a, a,..., a + Let a fo,,..., +, q whee is a oegative itege ad q, q,..., q + ae all odd positive iteges less tha Accodig to pigeohole piciple, because oly odd positive iteges less tha, two of the iteges q, q,..., q + must be equal Let q be the commo value of q i ad q, the, a i i q ad a q It follows that if i <, the a i divides a ; a q othewise a divides a i i q a i i 5 Geealized Pigeohole Piciple Example N/ N ( - ) + What is the least umbe of aea codes eeded to guaatee that the 5 millio phoes i a state ca be assiged distict 0-digit telephoe umbes? Assume that telephoe umbes ae of the fom NXX-NX X- XXXX, whee the fist thee digits fom the aea code, N epesets a digit fom to 9 iclusive, ad X epesets ay digit. Diffeet phoe umbes fo NXX-XXXX is 8 x 0 6 8,000,000 N 5,000,000, 8,000,000 At least 5,000,000 / 8,000,000 4 of them must have idetical phoe umbes Hece, at least fou aea codes ae equied 4 Applicatios: Subsequece Suppose that a, a,..., a N is a sequece of eal umbes. A subsequece of this sequece is a sequece of the fom a i, a i,..., a i m whee < i < i <... < i m < N 6

5 Applicatios: Subsequece Example Example: a, a,..., a 5 5, 8,, 3, 5, 3, is a subsequece? 8, is a subsequece?, 3, 5, 8 is a subsequece? a, a 4, a 5 a, a 5 a 3, a, a 4, a Applicatios: Subsequece Theoem Evey sequece of + distict eal umbes cotais a subsequece of legth + that is eithe stictly iceasig o stictly deceasig Example Give a sequece: 8,, 9,, 4, 6,, 0, 5, 7 0 tem 3 + What is the legth of the logest i / deceasig subsequeces? + 4 Iceasig seque ce, 4, 6,, 4, 6, 7, 4, 6, 0, 4, 5, 7 Deceasig seque ce, 9, 6,5 7 9 Applicatios: Subsequece A sequece is called stictly iceasig if each tem is lage tha the oe that pecedes it A sequece is called stictly deceasig if each tem is smalle tha the oe that pecedes it 8 Applicatios: Subsequece Poof Let a be a sequece of, a,..., a + distict + eal umbes Associate a odeed pai (i, d ) to the tem a, whee i is the legth of the logest iceasig subsequece statig at a d is the legth of the logest deceasig subsequece statig at a 5, 8,, 3, (i, d ) (, 3) (i 4, d 4 ) (, ) 0

6 Applicatios: Subsequece Poof Suppose o iceasig o deceasig subsequeces is loge tha i ad d ae both positive iteges less tha o equal to, fo,,..., + By the poduct ule, possible odeed pais fo (i, d ) By the pigeohole piciple two of + odeed pais ae equal Theefoe, thee exist tems a s ad a t, with s < t such that i s i t ad d s d t Applicatios: Ramsey Theoy Ramsey theoy, afte the Eglish mathematicia F. Ramsey, deals with the distibutio of subsets of elemets of sets Two people eithe fieds o eemies A B A B Fieds Ee mies Mutual Fie d/eem ies A B A B A B C D ae mutual fieds/ee mies C D C D 3 Applicatios: Subsequece Poof Thee exist tems a s ad a t, with s < t such that i s i t ad d s d t We will show that this is impossible, a s,, a t, 5, 8,, 3, (, 3) (, ) Because the tems of the sequece ae distict, eithe a s < a t o a s > a t If a s < a t, the, because i s i t, a iceasig subsequece of legth i t + ca be built statig at a s, by taig as followed by a iceasig subsequece of legth it begiig at a t This is a cotadictio Similaly, if a s > a t, it ca be show that d s must be geate tha d t, which is a cotadictio Applicatios: Ramsey Theoy Example Assume that i a goup of six people Show that thee ae eithe thee mutual fieds o thee mutual eemies i the goup A B C D E F 4

7 Applicatios: Ramsey Theoy Example Let A be oe of the six people Accodig to pigeohole piciple ( 5/ 3), A at least has thee fieds, o thee eemies Fome Case: suppose that B, C, ad D ae fieds If ay two of these thee people ae fieds, the these tw o ad A fom a goup of thee mutua l fieds Othew ise, B, C, ad D fom a set of thee mutual eemies Simila to the latte case A B C D E F 5 Applicatios: Ramsey Theoy 5 people caot guaatee havig 3 mutual fieds/eemies B A C E D 7 Applicatios: Ramsey Theoy Ramsey umbe R(m, ) The miimum umbe of people at a paty such that thee ae eithe m mutual fieds o mutual eemies, assumig that evey pai of people at the paty ae fieds o eemies m ad ae positive iteges geate tha o equal to Example What is R(3, 3)? Aswe should be 6 I a goup of five people whee evey two people ae fieds o eemies, thee may ot be thee mutual fieds o thee mutual eemies 6 Biomial Theoem Let x ad y be vaiables, ad let be a oegative itege The Biomial Theoem shows: x+ x y 0 ( y) 0 0 x y + x y x y 8

8 Biomial Theoem Example What is the coefficiet of x y 3 i the expasio of (x - 3y) 5? ( x + ( 3 y )) 5 5 ( x) ( 3 y) , ( ) ( 3 ) 3 5! 3!! 3 3 Biomial Theoem Coollay Let be a positive itege. The ( ) 0 0 Poof 0 0 ( ) ( ( ( ) + ) 0 0 It implies that ) Biomial Theoem Coollay Let be a oegative itege. The 0 Poof ( + ) 0 0 Biomial Theoem Coollay 3 Let be a oegative itege. The 0 3 Poof (3) ( + )

9 Pascal's Idetity ad Tiagle + C C - + C Poof Suppose T is a set cotaiig + elemets Let a be a elemet i T, ad let S T - {a} Thee ae + C subsets of T cotaiig elemets + C subsets cotais eithe + C a C - C - elemets of S ad a, o elemets of S ad ot a ( C - ) ( C ) Theefoe, + C C - + C T a S 34 Theoem elated to Biomial Coeff icie ts + m Vademode s Idetity m 0 Poof Suppose: m items i a fist set ad items i a secod set The total umbe of ways to pic elemets fom the uio of these sets is m+ C Aothe way is to pic elemets fom the fist set ad the - elemets fom the secod set, whee is a itege with C Thee ae m C C - w ays The efoe, m C - C m C 0 C m + m 0 m C C m C C 0 36 Pascal's Idetity ad Tiagle Pascal s Idetity Let ad be positive iteges with. The + + C 0 C C Pascal's tiagle A geometic aagemet of the biomial coefficiets i a tiagle biomial coefficiet is the sum of two adacet biomial coefficiets i the pevious ow 33 Theoem elated to Biomial Coeff icie ts Vademode s Idetity Theoem: Vademode s Idetity Let m,, ad be oegative iteges with ot exceedig eithe m o. The m + m 0 35

10 Theoem elated to Biomial Coeff icie ts Vademode s Idetity Coollay If is a oegative itege, the 0 Theoem elated to Biomial Coefficiets Theoem Let ad be oegative iteges with. The Theoem elated to Biomial Coeff icie ts Vademode s Idetity Poof Recall, 0 Theefoe, Theoem elated to Biomial Coefficiets Poof: Coside + C + couts the bit stigs of legth + cotaiig + oes cotai + s + bits Aothe coutig way is to coside the possible locatios, amed, of the fial should equal to +, +,..., o bits cotai s 40

11 Theoem elated to Biomial Coefficiets bits cotai s + + Coside the fist - bits I this - bits, thee should be s Thee ae - C ways Recall, By the chage of vaiables - 4