If the last tem in the binomial expansion of `(2^(1/3)-1/(sqrt(2)))^n i s(1/(3^(5/3)))^(log_3 8)` , then 5th term from the beginning is `210` b. `420` c. `105` d. none of these

Options

A. `""^(10)C_(6)`
B. `2""^(10)C_(4)`
C. `1/2cdot^(10)C_(4)`
D. None of the above

Explanation

Since, last term in the expansion of `(root(3)(2)-1/sqrt(2))^(n)`
`=(1/(3 cdot root(3)(9)))^(log_(3)8) rArr ""^(n) C_(n)cdot (-1/sqrt(2))^(n) = (1/(3cdot root(3)(9)))^(log_(3)8)`
`rArr (-1)^(n) cdot (1/2)^(n//2) = (1/(3^(5//3)))^(log_(3)8) = (3^(-5//3))^(log_(3)2^(3))`
`=3^(-5/3xx3xxlog_(3)2) = 3^(-5log_(3)2) = 3^(log_(3)2^(-5) = 2 ^(-5)=(1/2)^(5)`
`rArr (-1) ^(n) cdot (1/2)^(n//2) = (1/2)^(5)therefore n = 10`
Now 5th term from beginning `=""^(10) C_(4) (root(3)(2))^(6) (-1/sqrt(2))^(4)`
`=""^(10) C_(4) cdot 2^(2)cdot 1/2^(2) = ""^(10)C_(4)=""^(10)C_(6)`