Find the coefficient of $x^7$ in the expansion of $(1-x)^(-2)$ ?

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Answer 1 · 2017-10-15T06:34:42

The coefficient of $x^7$ is $8$

The Binomial expansion of $(1+x)^n$ is given by

$(1+x)^n=sum_(r=0)^(r=n)C_r^nx^r$ ,

where $C_r^n=(n!)/(r!(n-r)!)=(n(n-1)(n-2)...(n-r+1))/(1*2*3....*r)$

that is coefficients would be $1,n,(n(n-1))/(1*2),(n(n-1)(n-2))/(1*2*3),.....$

Hence $(1-x)^(-2)=sum_(r=0)^(r=n)C_r^(-2)(-x)^r$

i.e. $(1-x)^(-2)=1+((-2))/1(-x)+((-2)(-3))/(1*2)(-x)^2+((-2)(-3)(-4))/(1*2*3)(-x)^3+...$

and term containing $x^7$ woud be

$((-2)(-3)(-4)(-5)(-6)(-7)(-8))/(1*2*3*4*5*6*7)(-x)^7$

= $8x^7$

And the coefficient of $x^7$ is $8$

Find the coefficient of x^7x^7 in the expansion of (1-x)^(-2)(1-x)^(-2)?