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The coefficient of  x^n in the expansion of   (1+x)(1-x)^n is [AIEEE 2004]

 

Option: 1

(-1)^{n-1} n


Option: 2

(-1)^n(1-n)


Option: 3

(-1)^{n-1}(n-1)^2


Option: 4

(n-1)


Answers (1)

best_answer

Coefficient of  x^n in   (1+x)(1-x)^n=  Coefficient of x^n  in  (1-x)^n+ coefficient of   x^{n-1} \text { in }(1-x)^n \\

=   Coefficient of   x^n  in  \left[{ }^n C_n(-x)^n+x \cdot{ }^n C_{n-1}(-x)^{n-1}\right]=(-1)^n{ }^n C_n+(-1)^{n-1} \cdot{ }^n C_1=(-1)^n+(-1)^n \cdot(-n)=(-1)^n[1-n] .

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shivangi.bhatnagar

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