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\left(\;^{2 n} C_{0}\right)^{2}-\left(^{2 n} C_{1}\right)^{2}+\left(^{2 n} C_{2}\right)^{2}-\ldots+\left(^{2 n} C_{2 n}\right)^{2}  is equal to 

Option: 1

^{2 n} C_{n}


Option: 2

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


Option: 3

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


Option: 4

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


Answers (1)

best_answer

\begin{aligned}(x+1)^{2 n}=^{2 n} C_{0} x^{2 n}+^{2 n} C_{1} x^{2 n-1} &+^{2 n} C_{2} x^{2 n-2}\ldots+^{2 n} C_{2 n}\ldots \text { (i) } \\ \end{aligned}

and

\\ (1-x)^{2 n}=\;^{2 n} C_{0}-^{2 n} C_{1} x+^{2 n} C_{2} x^{2}-\ldots+^{2 n} C_{2 n} x^{2 n} \ldots \text { (ii) }\

\begin{array}{l}{\text { On multiplying Eqs. (i) and (ii), we get }} \\ {\left(1-x^{2}\right)^{2 n}=\left(^{2 n} C_{0}-^{2 n} C_{1} x+^{2 n} C_{2} x^{2}-\ldots+^{2 n} C_{2 n} x^{2 n}\right)} \\ {\qquad \qquad\qquad\qquad\qquad \times\left(^{2 n} C_{0} x^{2 n}+^{2 n} C_{1} x^{2 n-1}++\ldots+^{2 n} C_{2 n}\right)}\end{array}

\begin{array}{l}{\text { Now, coefficient of } x^{2 n} \text { in RHS }} \\ {\qquad \quad=\left(^{2 n} C_{0}\right)^{2}-\left(^{2 n} C_{1}\right)^{2}+\left(^{2 n} C_{2}\right)^{2}-\ldots+\left(^{2 n} C_{2 n}\right)^{2}}\end{array}

{\therefore \text { General term in LHS, } T_{r+1}=^{2 n} C_{r}\left(-x^{2}\right)^{r}}

To get coefficient of x2n in LHS, put r=n  

We get, T_{n+1}=(-1)^{n} \cdot^{2 n} C_{n}

Option B is correct

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manish

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