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The coefficient of x^(50) in (x+^(101)C(...

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

A

`4^(50)`

B

`2^(50)`

C

`2^(101)-1`

D

`2^(101)`

Text Solution

Verified by Experts

The correct Answer is:
A
`(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)`
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