Find the coefficient of `x^(4)` in the expansion of `(x//2- 3//x^(2))^(10)`.

Options

A. `(405)/(256)`
B. `(504)/(259)`
C. `(450)/(263)`
D. none of these

Answer

A

Explanation

Suppose `x^(4)` occours in `(r +1)^(th)` term.
We have ,
`T_(r+1) = ""^(10)C_(r) ((x)/(2))^(10-r) (-(3)/(x^(2)))^(3r) = ""^(10)C_(r) x^(10-3r)(-3)^(r) 2^(r-10)`
This will contain `x^(4)`,if
`therefore 10 - 3r = 4 rArr 3r = 6 rArr r =2`
So, `x^(4)` occours in 3rd and its coefficient is
` ""^(10)C_(r)xx(-3)^(2) xx2^(2-10) =""^(10)C_(2)xx(3^(2))/(2^(8)) = (5xx9xx3^(2))/(2^(8)) = (405)/(256)` .