The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!! We also define 0! Combinatoics- Pemutations and Combinations Suppose you ae asked to pick diffeent numbes between and. Thee ae 0 ways of doing this:,,,,,,,,,, The ode in which we pick the numbes is not impotant,,,, e.g.,, is the same as,,.,, This is called a combination.,, It is a selection without aangement.,, n Combinations use the notation n C o, whee you ae selecting components fom a total of n. Fomula n C n!! n!
Auchmuty High School Mathematics Depatment In the above example we ae selecting things fom. This is C o C!! 0 0!!!! 6. Lean how to calculate n! and n C on you calculato. If the ode (aangement) of the numbes is impotant this is a diffeent calculation. Suppose we ae selecting diffeent numbes fom whee the ode does matte. This time thee ae going to be moe possibilities.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,. and so on. Thee ae 60 possibilities altogethe. Think of it like this:- Fo the fist numbe thee ae choices,,, o. Fo the second numbe thee ae choices as you have used one numbe aleady. Fo the thid numbe thee ae choices as you have used numbes aleady. So in total you have 60 possibilities. is the same as doing!! This is known as a pemutation when aangement is impotant. It is denoted n P. Fomula n P n! n! In the above example!! 0 P 60.!!
Auchmuty High School Mathematics Depatment Pemutations ae not pat of the Advanced Highe couse but have been mentioned hee to fom a complete pictue and fo those who will study futhe mathematics. We will concentate on n C. Example people have to be selected fom 8 to fom a committee. How many ways ae thee to do this? 8 7 6 670 This is the same as calculating 8 7 6 8!.! But this includes all the possible aangements. Aangements don t matte hee so we need to divide by! as this is the numbe of ways things can be aanged. 8! So we have!!. It is easie to use ou fomula : Example How many ways can I place discs into empty boxes? C!! 0 0!!!! 6 NB This is also the same as placing empty boxes in. C!! 0 0!!!! 6 This implies that C C o. 8 C 8 8! 8! 00 6! 8!!! 06 Example How many diffeent ways can you place swimmes in 8 lanes? 8 C 8 8! 8! 00 6! 8!!! 60
This is the same as placing gaps in 8 lanes. 8 C 8 8! 8! 00 6! 8!!! 06 So 8 C 8 C o 8 8. Auchmuty High School Mathematics Depatment n n The geneal esult is n Poof n n! n n! n n! n!! n n! n! n!! Pascals Tiangle 6 0 0 Notice that the esults fom combinations occu in Pascal s tiangle. 0 0 0 0 0 0 etc.
Auchmuty High School Mathematics Depatment Fom the tiangle we can see anothe esult: ( n ) + (n ) = (n + ) Poof n n n! n!! n!! n! n! n!!!!! n n n! n! n! n!! n! n! n! n! n! n! n! n! n! n! n! n!! n! common denominato of n since n n n!!!! n as equied. Question fom the 00 pape equied a woking of this poof:- n n n Show that n n n! n!! n!! n! n! n! n n!!!!
Auchmuty High School Mathematics Depatment n! n! n! n!! n! common denominato of! n! n! n! n! n! n n! n! n! n! since n! n n! n! n n! n! n!! n! n!! n! n as equied. Questions Given that 0 (a) 6 0, 0 0 0 and 0 7 0 0 (b) (c) 8 wite down the value of Find (a) 8 (b) (c) 6 (d) Find anothe n C equivalent to (a) 6 (b) (c) 9 (d) 0 Wite down in n fom (a) 8 8 6 (b) (c) (d) 8 8 8 9 6
Auchmuty High School Mathematics Depatment Equations n Suppose we know that. Can we solve this fo n? n n!! n! n! n! n n! n n n n 0 n n0 0 n 6 n 0 n 6 o n n6, n n 66. What is the value of n? n 66 n! 66! n! n! n! n n! 66 n n 66 n n n n 0 n n 0 n o n n, n 7
Auchmuty High School Mathematics Depatment Solve fo n. n n 8 n n n n 8 using n! 8! n! n nn! n!! 8 n n 6 n n6 0 n 7 n8 0 n 7 o n 8 n7, n Questions n Find the value of n, n. (a) 0 n (b) 6 n (c) 0 n n Solve (a) (b) n n 66 n n (c) 90 The Binomial Theoem The Binomial Theoem helps us to multiply out backets which we would othewise have to complete longhand. 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 coefficients and compae with Pascal s Tiangle. 6 x y coefficients of x y coefficients of x y coefficients of 8
Auchmuty High School Mathematics Depatment n So the coefficients ae the same as o n C and ae known as the binomial coefficients. 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 0 0 0 0 In geneal n n n 0 n n n n n n n 0 n x y x y x y x y xy x y 0 n n fo x, y, n This is known as The Binomial Theoem. n n n n It can also be witten as x y x y fo n,. 0 n n The geneal tem of the expansion is given by x y. n You may choose to use Pascal s tiangle o AH level, Pascal s tiangle is usually sufficient. to find the coefficients it s up to you. At Examples x y x y x y x y x y x y x y 0 0 0 x x y 0x y 0x y x y y Be caeful when thee ae coefficients within the backet! 6 x y x x y x y x y y 6x x y x y 8xy y 9
x y x x y xy y Auchmuty High School Mathematics Depatment 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 Questions Expand using the binomial theoem: (a) x y 7 (b) a b 6 (c) x y 6 (d) a b (e) x y Examples involving negatives and factions Exta cae must be taken 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 x y 6 x 6x y x y 0x y x y 6x y y 6 6 79x 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
Auchmuty High School Mathematics Depatment 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 6 6 6 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 Questions Expand using the binomial theoem. (a) x y (b) 6x y (c) y x (d) a b 6 (e) a b 6 (f) a b (g) x y (h) 6 x x
Finding a Paticula Tem Auchmuty High School Mathematics Depatment You may be asked to find a paticula tem in an expansion o obtain its coefficient. This can be done by completing a whole expansion and picking out the equied tem but this can be time consuming and aithmetical eos ae moe likely to occu. n n n n It helps if you emembe the geneal fomula x y x y fo n,. 0 Examples Find the coefficient of the xy tem in the expansion of x y 7. Fo xy, n 7,, n. 7 The tem is x y x y coefficient Find the coefficient of the xy tem in the expansion of x y 6. Fo xy, n 6,, n. 6 The tem is x y 6x y 0x y coefficient 0 Find the tem independent of x in the expansion of Tem independent of x equies n 0,, n x x. 0 x x. 0 The tem is x x x 806 x
Auchmuty High School Mathematics Depatment Find the x tem in the expansion of x tem equies x x. n 6,, n 6 x 6 6x x 80x The tem is x Questions 6 x x. Find the coefficient of the Find the coefficient of the xy tem in the expansion of x y 6. x. 9 x tem in the expansion of Find the y tem in the expansion of y. y Find the tem independent of y in the expansion of Find the tem independent of a in the expansion of 8 y. y 9 a a. Witing down the Geneal Tem in an Expansion Remembe the geneal tem of the expansion of x y n n is given by x Examples Wite down and simplify the geneal tem in the expansion of x x. Hence o othewise obtain the tem in x. 0 x 0 0 x x 0 0 x 0 0 so tem is x x The th tem is given by x 0 0. n 0 y.
Auchmuty High School Mathematics Depatment Wite down and simplify the geneal tem in the expansion of Hence o othewise obtain the tem independent of x. 9 x x. The th tem is given by 9 9 0 9 x x 9 9 x x 9 9 9 x x 9 9 9 x 9 9 76. so tem is 6 6 Questions Wite down and simplify the geneal tem in the expansion of Hence o othewise obtain the tem in 0 x. Wite down and simplify the geneal tem in the expansion of Hence o othewise obtain the tem independent of x. 8 x x. x x. Applications of the Binomial Theoem We can use the binomial theoem to tackle othe types of poblems. Using the binomial theoem find 0 0 0 0. 0 0 6 0 0 0 0 0 0 008 000 00000 0 0000006 086 Using the binomial theoem find 06 0 06. 0 0 0 08 0 06 0 6
Expand x y x y Auchmuty High School Mathematics Depatment The tick hee is to notice it s a diffeence of squaes. x y x y x yx y x y Now use the binomial theoem. x x y x y y x x y x y y 6 6 Questions Calculate (a) 0 (b) 98 (c) 99 Expand the following (a) a b a b (b) a b a b (c) y y x x
Auchmuty High School Mathematics Depatment Past Pape Questions 00 A6 Expand 00 Q x x, x 0 and simplify as fa as possible. ( maks) Obtain the binomial expansion of a. ( maks) 007 Q Expess the binomial expansion of fo integes a, b, c, d and e. x x in the fom d e ax bx c x x ( maks) 008 Q8 Wite down and simplify the geneal tem in the expansion of x x. Hence o othewise, obtain the tem in x. (, maks) 009 Q8 (a) Wite down the binomial expansion of x. (b) Hence show that 09 is 0 909. (, maks) 0 00 Q Show that n n n whee the intege n is geate than o equal to. ( maks) 0 Q Use the binomial theoem to expand x and simplify you answe. ( maks) 0 Q Wite down and simplify the geneal tem in the expansion of Hence, o othewise, obtain the tem independent of x. 9 x x. (, maks) 6
Auchmuty High School Mathematics Depatment 0 Q Wite down the binomial expansion of x x and simplify you answe. ( maks) 0 Q Wite down and simplify the geneal tem in the expession Hence, o othewise, obtain the tem in x 0 Q Use the binomial theoem to expand and simplify 0 x x.. ( maks) Show that n n n x. ( maks) x, fo all integes, n, whee n. ( maks) 06 Q Wite down and simplify the geneal tem in the binomial expansion of x x. Hence, o othewise, find the tem in x 9. ( maks) 07 Q Wite down the binomial expansion of ( y y) and simplify you answe. ( maks) 7