Question 278 of 347MediumMCQ
The coefficient of xn in expansion of (1+x)(1−x)n is:
Solutions · 1
Official solution
Given \[\left( {1 + x} \right){\left( {1 - x} \right)^n}\]
= \[{\left( {1 - x} \right)^n}\] + \[x{\left( {1 - x} \right)^n}\]
General term of \[{\left( {1 - x} \right)^n}\] = \[{}^n{C_r}.{\left( { - 1} \right)^r}.{x^r}\]
\[\therefore\] Term containing xn in \[{\left( {1 - x} \right)^n}\] = \[{}^n{C_n}.{\left( { - 1} \right)^n}.{x^n}\]
So coefficient of xn in \[{\left( {1 - x} \right)^n}\] = \[{}^n{C_n}.{\left( { - 1} \right)^n}\]
General term of \[x{\left( {1 - x} \right)^n}\] = \[{}^n{C_r}.{\left( { - 1} \right)^r}.{x^{r + 1}}\]
\[\therefore\] Term containing xn in \[x{\left( {1 - x} \right)^n}\] = \[{}^n{C_{n - 1}}.{\left( { - 1} \right)^{n - 1}}.{x^n}\]
So coefficient of xn in \[x{\left( {1 - x} \right)^n}\] = \[{}^n{C_{n - 1}}.{\left( { - 1} \right)^{n - 1}}\]
\[\therefore\] coefficient of xn in \[{\left( {1 - x} \right)^n}\] + \[x{\left( {1 - x} \right)^n}\]
= \[{}^n{C_n}.{\left( { - 1} \right)^n}\] + \[{}^n{C_{n - 1}}.{\left( { - 1} \right)^{n - 1}}\]
= \[{\left( { - 1} \right)^{n - 1}}\left[ {{}^n{C_{n - 1}} - {}^n{C_n}} \right]\]
= \[{\left( { - 1} \right)^{n - 1}}\left[ {n - 1} \right]\]
= \[{\left( { - 1} \right)^n}\left[ {1 - n} \right]\]
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