查看完整版本 : binomial theorem 求高手入睇完答案都唔明.....

Jayw123 2018-11-30 10:38 AM

binomial theorem 求高手入睇完答案都唔明.....

唔明(n,n-i) 點黎[attach]9058023[/attach]

Jayw123 2018-11-30 10:55 AM

我GA 做法係咁但唔知點整走X^i

ppresent 2018-11-30 10:10 PM

向(1+x)^n*(1+x)^n入邊,所有x^n既coefficients都係源自左邊個term, (n,i)x^i 同右邊個term (n,n-i)x^n-i 既product
(右邊x既exponent必然是n-i才會令乘埋係x^n)
i.e. [img]https://latex.codecogs.com/gif.latex?%281+x%29%5En%3D1+nx+%20%7Bn%5Cchoose2%7D%20x%5E2%20+%20%7Bn%5Cchoose3%7Dx%5E3+...[/img]
[img]https://latex.codecogs.com/gif.latex?%281+x%29%5En%281+x%29%5En%3D%281+nx+%20%7Bn%5Cchoose2%7D%20x%5E2%20+%20%7Bn%5Cchoose3%7Dx%5E3+...%29%281+nx+%20%7Bn%5Cchoose2%7D%20x%5E2%20+%20%7Bn%5Cchoose3%7Dx%5E3+...%29[/img]
當中左邊的nx和右邊的[img]https://latex.codecogs.com/gif.latex?%7Bn%5Cchoose%7Bn-1%7D%7Dx%5E%7Bn-1%7D[/img]乘埋得出[img]https://latex.codecogs.com/gif.latex?n%5E2x%5E%7Bn%7D[/img]
亦有左邊的[img]https://latex.codecogs.com/gif.latex?%7Bn%5Cchoose%7B2%7D%7Dx%5E%7B2%7D[/img]和右邊的[img]https://latex.codecogs.com/gif.latex?%7Bn%5Cchoose%7Bn-2%7D%7Dx%5E%7Bn-2%7D[/img]乘埋得出[img]https://latex.codecogs.com/gif.latex?%7Bn%5Cchoose%7B2%7D%7D%5E2x%5E%7Bn%7D[/img]

而x^n的coefficient就是加哂呢d product terms既coefficient,  因此要loop哂由i=0 to i=n
所以x^n的coefficent就是 [img]https://latex.codecogs.com/gif.latex?%5Csum_%7Bi%3D0%7D%5E%7Bn%7D%7Bn%5Cchoose%20i%7D%7Bn%20%5Cchoose%20n-i%7D[/img]

ppresent 2018-11-30 10:12 PM

見圖

[[i] 本帖最後由 ppresent 於 2018-11-30 10:23 PM 編輯 [/i]]

Jayw123 2018-12-2 01:48 AM

CHING 想問你講GA左邊加邊係指緊咩? 係講緊(2n,n). 同埋想問係咪(1+X)^2N=(2N,N)?洗唔洗證埋

Jayw123 2018-12-2 01:50 AM

同埋唔明點得出(n,n-i)x^n-i

Jayw123 2018-12-2 03:42 AM

x^n ga coefficient 唔係(2n,n) 咩?

Jayw123 2018-12-2 03:50 AM

CHING SORRY 但我真係睇唔明左邊右邊係指緊咩同埋左邊nx 同右邊(n,n-1) 點突然出黎........

ppresent 2018-12-2 09:18 AM

[align=left]如圖[/align]

[[i] 本帖最後由 ppresent 於 2018-12-2 09:24 AM 編輯 [/i]]
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查看完整版本: binomial theorem 求高手入睇完答案都唔明.....