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Prove that the coefficient of x^r in the...

Prove that the coefficient of `x^r` in the expansion of `(1-2x)^(1//2)i s(2r)!//[2^r(r !)^2]dot`

Text Solution

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Coefficient of `x^(r )` is
`((-1/2)(-1/2-1)(-1/2-2)"....."(-1/2-r+1))/(r!)(-2)^(r)`
`= (1xx3xx5xx"..."xx(2r-1))/(r!)((-1)^(r)(-1)^(r)2^(r))/(r!)`
`= (1xx3xx5xx"..."xx(2r-1))/(r!)`
`= (1xx2xx3xx4xx5xx"...."xx(2r-1)(2r))/((2xx4xx6xx8xx"...."xx(2r))r!) = ((2r)!)/(2^(r)(r!)^(2))`
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