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

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-1} C_{n}+C_{2} \cdot^{2 n-2} C_{n}-C_{3} \cdot^{2 n-3} C_{n}+\ldots+(-1)^{n} C_{n} \cdot^{n} C_{n}

is equal to?

Option: 1

\frac{1}{n}


Option: 2

2^{-n}


Option: 3

2^n


Option: 4

1


Answers (1)

best_answer

Binomial Inside Binomial

C_{0} \cdot^{2 n} C_{n}-C_{1} \cdot^{2 n-1} C_{n}+C_{2} \cdot^{2 n-2} C_{n}-C_{3} \cdot^{2 n-3} C_{n}+\ldots+(-1)^{n} C_{n} \cdot^{n} C_{n}

= Coefficient of xn in 

\begin{array}{l}{\left[C_{0}(1+x)^{2 n}-C_{1}(1+x)^{2 n-1}+C_{2}(1+x)^{2 n-2}\right.} \\ {\left.\quad-C_{3}(1+x)^{2 n-3}+\ldots+(-1)^n .^{n} C_{n} \cdot(1+x)^{n}\right]}\end{array}

= Coefficient of xn in  

\begin{aligned}(1+x)^{n}\left[C_{0}(1+x)^{n}-\right.& C_{1}(1+x)^{n-1}+C_{2}(1+x)^{n-2} \\ &\left.-C_{3}(1+x)^{n-3}+\ldots+(-1)^n.^{n} C_{n} \cdot 1\right] \end{aligned}

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

= Coefficient of xn in (1+x)^{n} \cdot x^{n}

= Constant term in (1+x)^{n}=1

Option D is correct

Posted by

Gautam harsolia

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