Programme F 7 Binomials PROGRAMME F 7 BINOMIALS

Programme F 7: Binomials PROGRAMME F 7 BINOMIALS STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Binomial expansions The sigma notation The exponential number e STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Factorials Combinations Three properties of combinatorial coefficients STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Factorials If n is a natural number then the product of the successive natural numbers: 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 F 7: Binomials Factorials and combinations Combinations There are locations. different ways of arranging r different items in n different If the items are identical there are r! different ways of placing the identical items within one arrangement without making a new arrangement. So, there are locations. different ways of arranging r identical items in n different This denoted by the combinatorial coefficient STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Three properties of combinatorial coefficients STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Binomial expansions The sigma notation The exponential number e STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Binomial expansions Pascal’s triangle Binomial expansions The general term of the binomial expansion STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Binomial expansions 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 F 7: Binomials Binomial expansions 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 F 7: Binomials 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 expansions: STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Binomial expansions Notice that the coefficients in the expansions are the same as the numbers in Pascal’s triangle: STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Binomial expansions The power 4 expansion can be written as: or as: STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Binomial expansions The general power n expansion can be written as: This can be simplified to: STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Binomial expansions The general term of the binomial expansion The (r + 1)th term in the expansion of STROUD is given as: Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Binomial expansions The sigma notation The exponential number e STROUD Worked examples and exercises are in the text

Programme F 7: Binomials 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 F 7: Binomials 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: 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 F 7: Binomials The sigma notation General terms The sigma notation form of the binomial expansion is. STROUD Worked examples and exercises are in the text

Programme F 7: Binomials The sigma notation The sum of the first n natural numbers can be written as: STROUD Worked examples and exercises are in the text

Programme F 7: Binomials The sigma notation Rules for manipulating sums Rule 1: Constants can be factored out of the sum Rule 2: The sum of sums STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Factorials and combinations Binomial expansions The sigma notation The exponential number e STROUD Worked examples and exercises are in the text

Programme F 7: Binomials The exponential number e The binomial expansion of STROUD is given as: Worked examples and exercises are in the text

Programme F 7: Binomials The exponential number e The larger n becomes the smaller to 0. This fact is written as the limit of becomes – the closer its value becomes as is 0. Or, symbolically STROUD Worked examples and exercises are in the text

Programme F 7: Binomials The exponential number e Applying this to the binomial expansion of STROUD gives: Worked examples and exercises are in the text

Programme F 7: Binomials The exponential number e It can be shown that is a finite number whose decimal form is: 2. 7182818. . . This number, the exponential number, is denoted by e. STROUD Worked examples and exercises are in the text

Programme F 7: Binomials 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: STROUD Worked examples and exercises are in the text

Programme F 7: Binomials Learning outcomes üDefine n! and recognise that there are n! different combinations of n different items üEvaluate n! using a calculator and manipulate expressions involving factorials üRecognize that there are different combinations of r identical items in n locations üRecognize simple properties of combinatorial coefficients üConstruct Pascal’s triangle üWrite down the binomial expansion for natural number powers üObtain specific terms in the binomial expansion using the general term üUse the sigma notation üRecognize and reproduce the expansion for ex where e is the exponential number STROUD Worked examples and exercises are in the text