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  • If <img alt="(1+x)^{n}=C_{0}+C_{1} x \ldots\ldots +C_{n} x^{n}" src="https://learn.careers360.com/latex-image/?%281&plus;x%29%5E%7Bn%7D%3DC_%7B0%7D&plus;C_%7B1%7D%20x%20%5Cldots%5Cldot

If (1+x)^{n}=C_{0}+C_{1} x \ldots\ldots +C_{n} x^{n} then, C_{0} \cdot^{2 n} C_{n}-C_{1} \cdot^{2 n-2} C_{n}+C_{2} \cdot^{2 n-4} C_{n}-\ldots

is equal to?

Option: 1

2^{2n}


Option: 2

2^{n-1}


Option: 3

2^{n+1}


Option: 4

2^n


Answers (1)

best_answer

C_{0} \cdot^{2 n} C_{n}-C_{1} \cdot^{2 n-2} C_{n}+C_{2} \cdot^{2 n-4} C_{n}-\ldots

Coefficient of xr in \left[C_{0}(1+x)^{2 n}-C_{1}(1+x)^{2 n-2}+C_{2}(1+x)^{2 n-4}-\ldots\right]

Coefficient of xr in  \left.\left[C_{0}(1+x)^{2}\right]^{n}-C_{1}\left[(1+x)^{2}\right]^{n-1}+C_{2}\left[(1+x)^{2}\right]^{n-2}-\ldots\right]

Coefficient of xr in \left[(1+x)^{2}-1\right]^{n} \\

Coefficient of xr in {\left(2 x+x^{2}\right)^{n}}

Constant term in (2+x)^{n}=2^{n}

Option D is correct

Posted by

avinash.dongre

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