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

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

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

$$=\frac{1-x+xy}{1-x(1+y)}$$$$=(1-x+xy)\underset{k=0}{\sum ^{\mathrm{\infty }}}{x}^k{(1+y)}^k$$$$=(1-x+xy)\underset{k=0}{\sum ^{\mathrm{\infty }}}{x}^k\underset{r=0}{\sum ^k}{C}_r^k{y}^r$$$$coef.of{x}^n{y}^{m}is$$$$C_m^n-C_m^{n-1}+C_{m-1}^{n-1}=2C_{m-1}^{n-1}$$$$example:x^5{y}^4$$$$2C_3^4=8$$

Commented by Tawa11 last updated on 26/Sep/21

$$\mathrm{great}\mathrm{sir}$$

Commented by qaz last updated on 27/Sep/21

$$\mathrm{thanks}\mathrm{a}\mathrm{lot}.\mathrm{mr}\mathrm{W}$$

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