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PROGRAMME F9 BINOMIAL SERIES STROUD Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e STROUD Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e STROUD Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Factorials Combinations Three properties of combinatorial coefficients STROUD Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Factorials If n is a natural number then the product of the successive natural numbers: n (n 1) (n 2) ( )3 21 is called n-factorial and is denoted by the symbol n! In addition 0-factorial, 0!, is defined to be equal to 1. That is, 0! = 1 STROUD Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Combinations n! There are different ways of arranging r different items in n different locations. (n r )! If the items are identical there are r! different ways of placing the identical items within one arrangement without making a new arrangement. n! So, there are different ways of arranging r identical items in n different ( n r )! r ! locations. This denoted by the combinatorial coefficient nCr STROUD n! (n r )!r ! Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Three properties of combinatorial coefficients STROUD Cn nC0 1 (a) n (b) n (c) n Cnr nCr Cr nCr 1 n1Cr 1 Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Pascal’s triangle Binomial expansions The general term of the binomial expansion STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Pascal’s triangle The following triangular array of combinatorial coefficients can be constructed where the superscript to the left of each coefficient indicates the row number and the subscript to the right indicates the column number: STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Pascal’s triangle Evaluating the combinatorial coefficients gives a triangular array of numbers that is called Pascal’s triangle: STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Binomial expansions A binomial is a pair of numbers raised to a power. For natural number powers these can be expanded to give the appropriate binomial series: (a b)1 a b (a b)2 a2 2ab b2 (a b)3 a3 3a2b 3ab2 b3 (a b)4 a4 4a3b 6a2b2 4ab3 b4 STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Binomial expansions Notice that the coefficients in the expansions are the same as the numbers in Pascal’s triangle: (a b)1 a b (a b)2 a2 2ab b2 (a b)3 a3 3a2b 3ab2 b3 (a b)4 a4 4a3b 6a2b2 4ab3 b4 STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Binomial expansions The power 4 expansion can be written as: (a b)4 1a4b0 4a3b1 6a2b2 4a1b3 1a0b4 or as: (a b)4 4 C0a4b0 4C1a3b1 4C2a2b2 4C3a1b3 4C4a0b4 STROUD Worked examples and exercises are in the text Programme F9: Binomial series Binomial series Binomial expansions The general power n expansion can be written as: (a b)n n C0anb0 nC1an1b1 nC2an2b2 nCr anrbr nCna0bn This can be simplified to: (a b)n an nan1b STROUD n(n1) n2 2 n(n1)(n2) n3 3 a b a b 2! 3! bn Worked examples and exercises are in the text Programme F9: Binomial series Binomial series The general term of the binomial expansion The (r + 1)th term in the expansion of (a b)n is given as: Cr anrbr n STROUD n! a n r br (nr )!r ! Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e STROUD Worked examples and exercises are in the text Programme F9: Binomial series The sigma notation General terms The sum of the first n natural numbers Rules for manipulating sums STROUD Worked examples and exercises are in the text Programme F9: Binomial series The sigma notation General terms If a sequence of terms are added together: f(1) + f(2) + f(3) + . . . + f(r) + . . . + f(n) their sum can be written in a more convenient form using the sigma notation: n f (r ) r 1 The sum of terms of the form f(r) where r ranges in value from 1 to n. f(r) is referred to as a general term. STROUD Worked examples and exercises are in the text Programme F9: Binomial series The sigma notation General terms The sigma notation form of the binomial expansion is. n n nr r n n r r Cr a b where Cr a b is the general term r 1 STROUD Worked examples and exercises are in the text Programme F9: Binomial series The sigma notation The sum of the first n natural numbers The sum of the first n natural numbers can be written as: 1 2 3 STROUD n r n(n 1) 2 r 1 n Worked examples and exercises are in the text Programme F9: Binomial series The sigma notation Rules for manipulating sums Rule 1: Constants can be factored out of the sum n n kf (r) k f (r ) r 1 r 1 Rule 2: The sum of sums n n n { f (r) g (r)} f (r) g (r) r 1 r 1 r 1 STROUD Worked examples and exercises are in the text Programme F9: Binomial series Factorials and combinations Binomial series The sigma notation The exponential number e STROUD Worked examples and exercises are in the text Programme F9: Binomial series The exponential number e n 1 The binomial expansion of 1 is given as: n n 2 3 1 n(n 1) 1 1 1 n ( n 1)( n 2) 1 n 1 n n 2! n 3! n 11 (11/ n) (11/ n)(1 2/ n) 2! 3! STROUD 1 n 1 n n n Worked examples and exercises are in the text Programme F9: Binomial series The exponential number e 1 The larger n becomes the smaller becomes – the closer its value n becomes to 0. This fact is written as the limit of 1 as n is 0. n Or, symbolically STROUD 1 Lim 0 n n Worked examples and exercises are in the text Programme F9: Binomial series The exponential number e n 1 Applying this to the binomial expansion of 1 gives: n n 1 (10) (10)(10) Lim 1 11 3! n 2! n 1 1 1 1 0! 1! 2! 3! 1 r 0 r ! STROUD Worked examples and exercises are in the text Programme F9: Binomial series The exponential number e 1 It can be shown that is a finite number whose decimal form is: r 0 r ! 2.7182818 . . . This number, the exponential number, is denoted by e. STROUD Worked examples and exercises are in the text Programme F9: Binomial series The exponential number e It will be shown in Part II that there is a similar expansion for the exponential number raised to a variable power x, namely: 2 3 x x e 1 x 2! 3! x r x r! r x r 0 r ! STROUD Worked examples and exercises are in the text Programme F9: Binomial series Learning outcomes Define n! and recognise that there are n! different combinations of n different items Evaluate n! using a calculator and manipulate expressions involving factorials Recognize that there are different combinations of r identical items in n locations Recognize simple properties of combinatorial coefficients Construct Pascal’s triangle Write down the binomial expansion for natural number powers Obtain specific terms in the binomial expansion using the general term Use the sigma notation Recognize and reproduce the expansion for ex where e is the exponential number STROUD Worked examples and exercises are in the text