Question:
The coefficient of $x^n$ in expansion of $(1+ x)(1- x)^n$ is
The coefficient of $x^n$ in expansion of $(1+ x)(1- x)^n$ is
Updated On: Jul 5, 2022
- $(- 1)^{n-1} n $
- $(- 1)^n (1 -n)$
- $(-1)^{n-1} (n-1)^2$
- $(n - 1 ) $
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The Correct Option is B
Solution and Explanation
Coeff. of $x^{n}$ in $ \left(1+x\right)\left(1-x\right)^{n}$
= coeff of $x^n$ in
$ \left(1+x\right)\left(1 - {^{n}C_{1}}x + {^{n}C_{2}}x^{2} - ...+\left(-1\right)^{n} \times \, ^nC_{n} x^{n}\right) $
$= \left(-1\right)^{n} \times \,{^{n}C_{n} }+ \left(-1\right)^{n-1}\, {^{n}C_{n-1}} = \left(-1\right)^{n} + \left(-1\right)^{n-1} .n$
$ = \left(-1\right)^{n}\left(1-n\right)$
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Concepts Used:
Binomial Theorem
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is
Properties of Binomial Theorem
- The number of coefficients in the binomial expansion of (x + y)n is equal to (n + 1).
- There are (n+1) terms in the expansion of (x+y)n.
- The first and the last terms are xn and yn respectively.
- From the beginning of the expansion, the powers of x, decrease from n up to 0, and the powers of a, increase from 0 up to n.
- The binomial coefficients in the expansion are arranged in an array, which is called Pascal's triangle. This pattern developed is summed up by the binomial theorem formula.