If the number of terms in the expansion of `(1-2/x+4/(x^2))^n , x!=0,` is 28, then the sum of the coefficients of all the terms in this expansion, is :

Options

A. 64
B. 2187
C. 243
D. 729

Answer

D

Explanation

Clearly, number of terms in the expansion of
`(1 - (2)/(x) + (4)/(x^(2)))^(n)` is `((n + 2)(n + 1))/(2)` or `.^(n + 2)C_(2)`
assuming `(1)/(x)` and `(1)/(x^(2))` distinct]
`:.((n+2) (n + 1))/(2) = 28`
`implies (n + 2) (n + 1) = 56 = (6 + 1) (6 + 1) implies n = 6`
Hence, sum of coefficients `= (1 - 2 + 4)^(6) = 3^(6) = 729`
Note As `(1)/(x)` and `(1)/(x^(2))` are functions of same variables therefore number of dissimilar terms will be `2n + 1`, i.e., odd, which is not possible, Hence it contains error.