The Binomial Theorem 5! ! ! 3 2 1! 1. Factorials
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- Blaise Garrett
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1 The Biomial Theoem Factoials The calculatios,, 6 etc. ofte appea i mathematics. They ae called factoials ad have bee give the otatio!. e.g. 6! 6!!!!! We also defie 0! Combiatoics- Pemutatios ad Combiatios Suppose you ae asked to pick diffeet umbes betwee ad. Thee ae 0 ways of doig this:,,,,,,,,,, The ode i which we pick the umbes is ot impotat,,,, e.g.,, is the same as,,.,, This is called a combiatio.,, It is a selectio without aagemet.,, Combiatios use the otatio C o, whee you ae selectig compoets fom a total of. Fomula C!!!
2 I the above example we ae selectig thigs fom. This is C o C!! 0 0!!!! 6. Lea how to calculate! ad C o you calculato. If the ode (aagemet) of the umbes is impotat this is a diffeet calculatio. Suppose we ae selectig diffeet umbes fom whee the ode does matte. This time thee ae goig to be moe possibilities.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,. ad so o. Thee ae 60 possibilities altogethe. Thik of it like this:- Fo the fist umbe thee ae choices,,, o. Fo the secod umbe thee ae choices as you have used oe umbe aleady. Fo the thid umbe thee ae choices as you have used umbes aleady. So i total you have 60 possibilities. is the same as doig!! This is kow as a pemutatio whe aagemet is impotat. It is deoted P. Fomula P!! I the above example!! 0 P 60.!!
3 Pemutatios ae ot pat of the Advaced Highe couse but have bee metioed hee to fom a complete pictue ad fo those who will study futhe mathematics. We will cocetate o C. Example people have to be selected fom 8 to fom a committee. How may ways ae thee to do this? This is the same as calculatig !.! But this icludes all the possible aagemets. Aagemets do t matte hee so we eed to divide by! as this is the umbe of ways thigs ca be aaged. 8! So we have!!. It is easie to use ou fomula : Example 8 C 8 8! 8! 00 6! 8!!! 06 How may ways ca I place discs ito empty boxes? C!! 0 0!!!! 6 NB This is also the same as placig empty boxes i. C!! 0 0!!!! 6 This implies that C C o. Example How may diffeet ways ca you place swimmes i 8 laes? 8 C 8 8! 8! 00 6! 8!!! 60 This is the same as placig gaps i 8 laes.
4 8 C 8 8! 8! 00 6! 8!!! 06 C So 8 8 C o 8 8. The geeal esult is Poof!!!!!!!!! Pascals Tiagle Notice that the esults fom combiatios occu i Pascal s tiagle etc. Fom the tiagle we ca see aothe esult:
5 ( ) + ( ) = ( + ) Poof!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! commo deomiato of sice!!!! as equied. Questio fom the 00 pape equied a wokig of this poof:- Show that!!!!!!!!!!!!!!!!!! commo deomiato of!!
6 !!!!!!!!!!!!!!!!! as equied. sice!! Questios Give that 0 (a) 6 0, ad (b) (c) 8 wite dow the value of Fid (a) 8 (b) (c) 6 (d) Fid aothe C equivalet to (a) 6 (b) (c) 9 (d) 0 Wite dow i fom (a) (b) (c) (d) Equatios 6
7 Suppose we kow that. Ca we solve this fo?!!!!!! o 6, 66. What is the value of? 66! 66!!!!! o, Solve fo. 7
8 8 8 usig! 8!!!!! o 8 7, Questios Fid the value of,. (a) 0 (b) 6 (c) 0 Solve (a) (b) 66 (c) 90 The Biomial Theoem The Biomial Theoem helps us to multiply out backets which we would othewise have to complete loghad. x y x xy y x y x y x xy y x x y xy y x y x y x x y xy y x x y 6x y xy y Look at the coefficiets ad compae with Pascal s Tiagle. 6 x y coefficiets of x y coefficiets of x y coefficiets of 8
9 So the coefficiets ae the same as o C ad ae kow as the biomial coefficiets. x y x y x y x y xy x y 0 x y x y x y x y x y x y x y 0 So I geeal 0 0 x y x y x y x y xy x y 0 fo x, y, This is kow as The Biomial Theoem. It ca also be witte as x y x y fo,. 0 The geeal tem of the expasio is give by x y. You may choose to use Pascal s tiagle o AH level, Pascal s tiagle is usually sufficiet. to fid the coefficiets it s up to you. At Examples x y x y x y x y x y x y x y x x y 0x y 0x y x y y Be caeful whe thee ae coefficiets withi the backet! 6 x y x x y x y x y y 6x x y x y 8xy y 9
10 x y x x y xy y x 9x y 7xy 7y a b a a b 0a b 0a b ab b 0a 80a b 760a b 0a b 60ab b Questios Expad usig the biomial theoem: (a) x y 7 (b) a b 6 (c) x y 6 (d) a b (e) x y Examples ivolvig egatives ad factios Exta cae must be take hee! x y x x y 6x y xy y x x y 6x y xy y x y x x y xy y 8x x y 6xy y 6 x y x 6x y x y 0x y x y 6x y y x 96x y 860x y 0x y 60x y 76xy 6y 6 6 x x x 6x x y y y y y x 6x x x y y y y 0
11 x x x 0x 0x x y y y y y y x 0x 0x x x y y 8y 6y y x x x x x y y y 6y y 6 x y x 6x x 0x x 6x y y y y y y 6 76x 60x 0x 860x 96x 79 6x 6 y y y y y y 7 x x x x 0x 0x x x x x x x 80x 080x 70x 0x x x x x x x 70 0 x x x x 80x 080x 8 x y x x y 6x y x y y x x y x y 08x y 8y 8 6 Questios Expad usig the biomial theoem. (a) x y (b) 6x y (c) y x (d) a b 6 (d) a b 6 (e) a b (f) x y (g) 6 x x
12 Fidig a Paticula Tem You may be asked to fid a paticula tem i a expasio o obtai its coefficiet. This ca be doe by completig a whole expasio ad pickig out the equied tem but this ca be time cosumig ad aithmetical eos ae moe likely to occu. It helps if you emembe the geeal fomula x y x y fo,. 0 Examples Fid the coefficiet of the xy tem i the expasio of 7 x y. Fo xy, 7,,. 7 The tem is x y x y coefficiet Fid the coefficiet of the xy tem i the expasio of 6 x y. Fo xy, 6,,. 6 The tem is x y 6x y 0x y coefficiet 0 Fid the tem idepedet of x i the expasio of Tem idepedet of x equies 0,, x x. 0 x x. The tem is 0 x x
13 x x 806 Fid the x tem i the expasio of x tem equies x x. 6,, 6 x 6 6x x 80x The tem is x Questios 6 x x. Fid the coefficiet of the xy tem i the expasio of 6 x y. Fid the coefficiet of the 9 x tem i the expasio of x. Fid the y tem i the expasio of y. y Fid the tem idepedet of y i the expasio of 8 y. y Fid the tem idepedet of a i the expasio of 9 a a. Witig dow the Geeal Tem i a Expasio Remembe the geeal tem of the expasio of x y is give by x Examples 0 Wite dow ad simplify the geeal tem i the expasio of x x. Hece o othewise obtai the tem i x. 0 The th tem is give by 0 x x 0 0 x x y.
14 0 0 x 0 0 so tem is x x 0 Wite dow ad simplify the geeal tem i the expasio of Hece o othewise obtai the tem idepedet of x.. 9 x x. The th tem is give by 9 x x 9 9 x x x x x so tem is 6 6 Questios Wite dow ad simplify the geeal tem i the expasio of 0 Hece o othewise obtai the tem i x. Wite dow ad simplify the geeal tem i the expasio of Hece o othewise obtai the tem idepedet of x. 8 x x. x x. Applicatios of the Biomial Theoem We ca use the biomial theoem to tackle othe types of poblems. Usig the biomial theoem fid Usig the biomial theoem fid
15 Expad x y x y The tick hee is to otice it s a diffeece of squaes. x y x y x yx y x y Now use the biomial theoem. x x y x y y x x y x y y 6 6 Questios Calculate (a) 0 (b) 98 (c) 99 Expad the followig (a) a b a b (b) a b a b (c) y y x x Past Pape Questios 00 Expad x x, x 0 ad simplify as fa as possible. ( maks) 00 Obtai the biomial expasio of a. ( maks) 007 Expess the biomial expasio of fo iteges a, b, c, d ad e. x x i the fom d e ax bx c x x ( maks) 008
16 Wite dow ad simplify the geeal tem i the expasio of x x. Hece o othewise, obtai the tem i x. (, maks) 009 (a) Wite dow the biomial expasio of x. (b) Hece show that 09 is (, maks) 0 00 Show that whee the itege is geate tha o equal to. ( maks) 0 Use the biomial theoem to expad x ad simplify you aswe. ( maks) 0 Wite dow ad simplify the geeal tem i the expasio of Hece, o othewise, obtai the tem idepedet of x. 9 x x. (, maks) 0 Wite dow the biomial expasio of maks) x x ad simplify you aswe. ( 0 Wite dow ad simplify the geeal tem i the expessio Hece, o othewise, obtai the tem i x 0 x x.. ( maks) 6
17 0 Use the biomial theoem to expad ad simplify Show that x. ( maks) x, fo all iteges,, whee. ( maks) 7
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