BINOMIAL THEOREM. 1. Introduction. 2. The Binomial Coefficients. ( x + 1), we get. and. When we expand

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1 BINOMIAL THEOREM Itoductio Whe we epad ( + ) ad ( + ), we get ad ( + ) = ( + )( + ) = = + + ( + ) = ( + )( + ) = ( + )( + + ) = = espectively Howeve, whe we ty to epad ( + ) ad ( + ), we fid that this method becomes vey tedious Futhemoe, how about ( + ), whe is a abitay positive itege? The biomial theoem helps us to epad these ad othe epessios moe easily ad coveietly Hee we shall go ove the basics of the biomial theoem, as well as a umbe of techiques i which the biomial theoem helps us to solve a wide age of poblems The Biomial oefficiets Defiitio (Biomial coefficiet) The biomial coefficiets ae the coefficiets i the epasio of ( + ) The coefficiet of i ( + ) is deoted as, o (Hee we shall be usig the fist otatio thoughout) The Pascal tiagle lists out all the biomial coefficiets, as show i Figue Page of

2 th ow st ow d ow d ow 4th ow th ow 5 5 6th ow th ow Figue : The Pascal tiagle The th ow is the coefficiet i the epasio of ( + ), o simply Fom the fist ow owads, the th ow cotais the + coefficiets i the epasio of ( + ) As you ca pobably obseve, thee ae some itiguig popeties i the Pascal tiagle: The tiagle is symmetic about the middle of each ow, ie = Ecept fo the leftmost ad ightmost umbes i each ow, evey umbe is the sum of the two umbes diectly above it, ie = + I evey ow, the umbes fist icease fom to a maimum value, the decease back to The secod umbe (as well as the secod last umbe) i the th ow is, ie = = Eample Usig the Pascal tiagle, epad 5 ( + ) ad 7 ( + ) Solutio Fom the Pascal tiagle, the tems i the 5th ow ae, 5,,,, 5 ad Thus we have ( ) = Similaly, by efeig to the 7th ow of the Pascal tiagle, we have ( ) = Page of

3 The Biomial Theoem We ae goig to fid a eplicit fomula fo computig the umbes i the Pascal tiagle, so that we ca cay out epasios coveietly without always efeig to the tiagle as well as eplai the iteestig popeties we obseved Accodig to defiitio, ( + ) = () Puttig =, we get =, as we ca see fom the Pascal tiagle The othe tems ca also be computed by successive diffeetiatio of the above fomula, as we illustate below Diffeetiatig () oce, we have ( + ) = + ( ) + ( ) Puttig = agai, we have = Similaly, by diffeetiatig () times, we have ( )( ) ( + )( + ) ( )( ) ( ) ( ) Puttig =, we get = ( )( ) ( + ) = + To make the fomula simple, we itoduce the factoial symbol Defiitio (Factoial) Fo positive itege, we defie the factoial of, deoted as!, by We also defie! =! = ( ) ( ) With this otatio i had, we ca give the fomula fo the biomial coefficiets i a ice fom: Page of

4 Theoem The biomial coefficiet is give by! =!( )! I combiatoics, deotes the umbe of ways of choosig objects fom objects The coefficiet of i ( + ) is ad this has its combiatoial meaig The tem comes by choosig s fom tems of ( + ) Thus thee ae tems ad hece its coefficiet is Although we have till ow oly discussed the epasio of ( + ), but fo the geeal epessio ( a+ b) it just takes a few moe steps Theoem (Biomial Theoem) ( a+ b) = a + a b+ a b + + ab + b ( a b) = a a b+ a b + ( ) ab + ( ) b Poof: We have ad so ( a+ b) = b + b a a a a a a a + = b b b b b b osequetly, ( a+ b) a a a a a = b b b b b b = b a b + a b + a b + + a b + ab + = a + a b+ a b + + ab as desied Fo the epasio of ( a b ) + ab + b, we simply apply the above esult with b eplaced by b QED Page 4 of

5 Eample Epad ( ) i ascedig powes of, up to the tem cotaiig Solutio ( ) = ( + ) = ( ) + ( ) + ( ) + ( ) ( 5) 58 ( 8) = = + + Eample Fid the coefficiet of i ( )( ) Solutio Sice 5 ( ) = 5( ) + (4 ) (8 ) + = the tem i (+ + )( ) is give by the sum of ( 8 ), So the coefficiet of is = 6 (4 ) ad ( ) Eample Fid the costat tem i the epasio of ( ) 6 Solutio I the epasio of ( ), i descedig powes of, the ( + ) st tem is ( ) = ( ) = ( ) 6 Fo the costat tem, the powe of is Thus we have = ad thus = 4 The the costat tem is ( ) = 5 6 = Page 5 of

6 Eample 4 Pove that + a+ a a a + Solutio The iequality is equivalet to ( + )!! a ( a+ )!( a)! a!( a)! a+ Afte simplificatio, this becomes + ( a)( a + ) Usig the AM-GM iequality, we have + ( a) + ( a+ ) = ( a)( a+ ) ad the oigial iequality is poved Eample 5 Fid the maimum coefficiet i the epasio of ( + 5) without actual epasio Solutio Note that the -th tem i the epasio is ( ) (5) We coside the atio of the ( + ) st tem to the -th tem, ie 5 = To see which coefficiet is maimum, we oly eed to kow whe the atio is geate tha ad whe it is smalle tha So we set > > = 8 8 Theefoe, the coefficiets icease at fist ad each a maimum whe = 4 So, the maimum coefficiet is = Page 6 of

7 4 Applicatios of the Biomial Theoem The obvious applicatio of the biomial theoem is to help us to epad algebaic epessios easily ad coveietly Othe tha that, the biomial theoem also helps us with simple umeical estimatios We illustate this with a couple of eamples Eample 4 A ma put $ ito a bak at a iteest ate of % pe aum, compouded mothly How much iteest ca he get i 5 moths, coect to the eaest dolla? Solutio Of couse with a calculato i had this will be of o difficulty But without it we ca still estimate the aswe usig the biomial theoem Afte 6 moths, the amout will be 5 () Usig the biomial theoem, 5 5 () = ( + ) = + 5() + () + () + 5() + () 4 5 Sice we oly equie the aswe to be accuate to the eaest dolla, we take oly the fist thee tems ad igoe the est We have 5 () = 5 Thus the amout is appoimately $5 ad the iteest is appoimately $5 Eample 4 Estimate the value of 6 (9) coect to decimal places Solutio Agai with a calculato this will just take a few secods But without it let us ty to epad We have ( + + ) 6 = [ + ( + ) ] 6 Puttig =, we get ( ) = + 6 ( + ) + 5 ( + ) + ( + ) + = (9) Page 7 of

8 5 Techiques fo Dealig with Biomial oefficiets Although we defied the biomial coefficiets as coefficiets i the biomial epasio of ( + ), we have emaked that these coefficiets have a vey ice combiatoial itepetatio ad the coefficiets themselves have may ice popeties I this sectio we shall demostate vaious techiques i wokig with biomial coefficiets I Sectio, we woked out a eplicit fomula fo the biomial coefficiets by diffeetiatio ad substitutio of the vaiable by cetai umbes It tus out that these ae impotat techiques that ca help us to deive othe popeties coceig the biomial coefficiets as well Recall that, accodig to ou defiitio, By puttig =, we get Similaly, by puttig =, we get ( + ) = = () ( ) = + + Addig ad subtactig these two equalities espectively ad dividig by, we get = = Now itegatig () (we use defiite itegal hee so that we ca get id of the costat of itegatio), we have Puttig =, we have ( + ) = ( ) d d + + ( + ) = = Itegatig oce moe, we have + ( + ) = ( + )( + ) + 4 ( + )( + ) 4 + Page 8 of

9 Puttig = agai, we see that = ( ) ( )( ) m m Now coside the idetity ( + ) ( + ) ( + ) + We shall compae the coefficiet of (assumig m ) o both sides O the left had side, we have ( )( ) m m m m ( + ) ( + ) = so that the coefficiet of is m m m m O the ight had side, the coefficiet of is simply m+ So we have the equality = + () m m m m m m m m Note that the idetity + + = (which we stated i Sectio without poof) ca be obtaied usig the above method by settig = It ca also be poved by applyig the fomula fo biomial coefficiets diectly, ad we leave it as a eecise Now settig = m= i (), we have = Sice =, this ca also be epessed as ( ) ( ) ( ) ( ) = So fa we have see some idetities ivolvig the biomial coefficiets We iclude below a few moe eamples which establish futhe popeties usig these idetities Eample 5 Let ad be iteges with Fid = Hece fid = Solutio We have Page 9 of

10 ! =!( )! = = = (! ) ( ) ( ) = =!! (! ) ( ) ( ( )) = =!! = = O the othe had, ( ) = = = = = = + ( ) = = = + = = (( ) ) = + ( ) = + Eample 5 Let a, b, be iteges such that a b Usig the idetity a + b a b a b a b = + + +, pove that a b ( a+ b ) a b Solutio By the auchy-schwaz iequality, we have a+ b a b a b a b ( ) = ( ) a a a b b b ( ) + ( ) + + ( ) ( ) + ( ) + + ( ) a a a b b b ( ) ( ) ( a ) ( ) ( ) ( b ) = a b a b Page of

11 Eample 5 Let ad be iteges with Pove that = Solutio Usig the idetity = + epeatedly, we have + + = = = = + = = The Biomial Theoem fo Negative ad No-Itegal Idices So fa we have bee cosideig the biomial theoem i cases whee the powe of the epasio is a o-egative itege I this sectio we biefly teat the cases whee the powe is a egative itege o a abitay eal umbe To begi ou discussio we fist eted the defiitio of the biomial coefficiets to cases whee the uppe ide is ot ecessaily a o-egative itege Defiitio 6 (Biomial coefficiet) Fo abitay eal umbe ad positive itege, we defie = ad ( ) ( ) ( + ) =! Note that this is the same fomula as we deived i Sectio We simply eted it to cases whee is ot ecessaily a o-egative itege To eted the biomial theoem to egative ad o-itegal idices, ecall that Page of

12 ( + ) = = Sice = whe > (eecise), this ca be witte as ( + ) = Whe is ot a o-egative itege, the above fomula gives a ifiite seies fo the epasio of ( + ) It ca be poved that the seies coveges fo < ad it ideed coveges to the epessio o the left had side = Fo istace, whe =, we have ( ) + = + + We kow that the (geometic) seies o the ight coveges wheeve <, ad the fomula o the left epesets the sum to ifiity 7 Eecises Epad the followig usig the biomial theoem (a) 4 ( ) (b) 6 Fid the coefficiet of i the epasio of (a) ( ) ( ) 4 + (b) 4 Pove that the costat tem i the epasio of multiple of 5, ad the costat tem is eve egative 4 is o-zeo if ad oly if is a a b 4 Fid all positive iteges a, b so that i the epessio ( + ) + ( + ), the coefficiets of ad ae equal Page of

13 5 Let ad be iteges, < Show that = 6 Pove that fo all positive iteges, ( ) = + 7 Eplai equatio () i Sectio 5 usig the combiatoial itepetatio of the biomial coefficiets 8 Pove the idetity + = + by usig the fomula fo the biomial coefficiets diectly m m m 9 Re-do Eample 5 by diffeetiatig equatio () i Sectio Fid the sum (Hit: oside a pimitive cube oot of uity, ie a comple umbe ω fo which ω ω + + = ad ω ) Let,, k be iteges such that k (a) Pove that k k k = (b) Give a combiatoial itepetatio of the equality i (a) The sequece of atala umbes,,, 5, 4, 4, is give by the fomula = =,,, + ( ) Pove that = fo all positive iteges + Let be a fied positive itege Fid, i tems of, the umbe of -digit positive iteges with the followig two popeties: (a) Each digit is eithe, o (b) The umbe of s is eve Page of