The coefficient of `x^(50)` in `(x+^(101)C_(0))(x+^(101)C_(1)).....(x+^(101)C_(50))` is

Options

A. `4^(50)`
B. `2^(50)`
C. `2^(101)-1`
D. `2^(101)`

Answer

A

Explanation

`(a)` `(x+^(101)C_(0))(x+^(101)C_(0))....(x+^(101)C_(50))` contains `51` linear factors.
`:.x^(50)` is obtained by multiplying `x's` from any `50` factors and the constant term from the remaining one factor.
Hence the coefficient of `x^(50)=S=^(101)C_(0)+^(101)C_(1)+...+^(101)C_(50)`
`"^(101)C_(51)=^(101)C_(50)`, `.^(101)C_(101)=^(101)C_(0)` etc.
`:.2S=^(101)C_(0)+^(101)C_(1)+...+^(101)C_(50)+^(101)C_(51)+..+^(101)C_(101)=2^(101)`
`impliesS=2^(100)implies4^(50)`