How-to-extract-the-coefficient-of-term-x-n-y-m-in-1-yx-1-x-1-yx-1-x-1-

Question Number 155189 by qaz last updated on 27/Sep/21

How to extract the coefficient of term “ x^{n} y^{m}^{″} in (1 + \frac{yx}{1 - x}) {(1 - \frac{yx}{1 - x})}^{- 1} ?

Answered by mr W last updated on 26/Sep/21

= \frac{1 - x + x y}{1 - x (1 + y)}

= (1 - x + x y) \underset{k = 0}{\sum^{\infty}} x^{k} {(1 + y)}^{k}

= (1 - x + x y) \underset{k = 0}{\sum^{\infty}} x^{k} \underset{r = 0}{\sum^{k}} C_{r}^{k} y^{r}

c o e f . o f x^{n} y^{m} i s

C_{m}^{n} - C_{m}^{n - 1} + C_{m - 1}^{n - 1} = 2 C_{m - 1}^{n - 1}

e x a m p l e : x^{5} y^{4}

2 C_{3}^{4} = 8

Commented by Tawa11 last updated on 26/Sep/21

great sir

Commented by qaz last updated on 27/Sep/21

thanks a lot . mr W

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