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Question: The coefficient of \(x^{n}\)in the expansion of \((1 + x)(1 - x)^{n}\) is...

The coefficient of xnx^{n}in the expansion of (1+x)(1−x)n(1 + x)(1 - x)^{n} is

A

(−1)n−1n( - 1)^{n - 1}n

B

(−1)n(1−n)( - 1)^{n}(1 - n)

C

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

D

(n−1)(n - 1)

Answer

(−1)n(1−n)( - 1)^{n}(1 - n)

Explanation

Solution

Coefficient of xnx^{n} in a0+a2+a4+......+a2na_{0} + a_{2} + a_{4} + ...... + a_{2n} = Coefficient of xnx^{n} in (1−x)n+(1 - x)^{n} + coefficient of xn−1in (1−x)nx^{n - 1}\text{in }(1 - x)^{n}

= Coefficient of xnx^{n} in [n⥂Cn(−x)n+x.n⥂Cn−1(−x)n−1]\lbrack^{n} ⥂ C_{n}( - x)^{n} + x.^{n} ⥂ C_{n - 1}( - x)^{n - 1}\rbrack

= (−1)nn⥂Cn+(−1)n−1.n⥂C1( - 1{)^{n}}^{n} ⥂ C_{n} + ( - 1)^{n - 1}.^{n} ⥂ C_{1} = (−1)n+(−1)n.(−n)=(−1)n[1−n]( - 1)^{n} + ( - 1)^{n}.( - n) = ( - 1)^{n}\lbrack 1 - n\rbrack.