Find the coefficient of `x^(6)` in the expansion of `(2x^(3)-(1)/(3x^(3)))^(10)`

Answer

N/a

Explanation

Let (r+1)th terms in the expansion of `(2x^(3)-(1)/(3x^(3)))^(10) " is " T_(r+1)`.
`T_(r+1)=^(10)C_(r)(2x^(3))^(10-r)(-(1)/(3x^(3)))^(r)`
`=^(10)C_(r).2^(10-r).x^(30-3r).((-1)^r)/(3^(r).x^(3r))`
`=^(10)C_(r).(2^(10-r).(-1))/(3^(r)).x^(30-6r)`
Let 30 -6r =6
`rArr " "6r =24`
`rArr " "r=4`
`:.` coefficient of `x^(6)=^(10)C_(4).(2^(10-4)(-1)^(4))/(3^(4))`
`(|ul10)/(|ul6|ul4)xx(64)/(81)`
`(7xx8xx9xx10)/(24)xx(64)/(81)`
`(4480)/(27).`