The sum of the coefficients of even power of `x` in the expansion of `(1+x+x^2+x^3)^5`is a.`256` b. `128` c. `512` d. `64`

Options

A. 256
B. 128
C. 512
D. 64

Answer

C

Explanation

`(1+x+x^(2)+x^(3))^(5) = a_(0) + a_(1)x + a_(2)x^(2) + a_(3)x^(3) + "….." + a_(15)x^(15)`
Putting `x = 1` and `x = - 1` alternatively, we have,
`{:(a_(0)+a_(1)+a_(2)+a_(3)+"....."+a_(15) = 4^(5) " "(1)),(a_(0)-a_(1)+a_(2)-a_(3)+"....."-a_(15) = 0 " "(2)):}`
Addiing (1) and (2), we have
`(2_(0) + a_(2) + a_(4) + "......"+a_(14)) = 4^(5)`
or `a_(0) + a_(2) + a_(4) + "....." + a_(14) = 2^(9) = 512`