If (1+x)n=C0+C1x+C2x2+…+Cnxn  then C02+C12+C22+C32+…+Cn2 is equal to

# then ${\mathrm{C}}_{0}^{2}+{\mathrm{C}}_{1}^{2}+{\mathrm{C}}_{2}^{2}+{\mathrm{C}}_{3}^{2}+\dots +{\mathrm{C}}_{\mathrm{n}}^{2}$ is equal to

1. A

$\frac{\mathrm{n}!}{\mathrm{n}!\mathrm{n}!}$

2. B

$\frac{\left(2\mathrm{n}\right)!}{\mathrm{n}!\mathrm{n}!}$

3. C

$\frac{\left(2\mathrm{n}\right)!}{\mathrm{n}!}$

4. D

None of these

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### Solution:

we have ,$\left(1+\mathrm{x}{\right)}^{\mathrm{n}}={\mathrm{C}}_{0}+{\mathrm{C}}_{1}\mathrm{x}+{\mathrm{C}}_{2}{\mathrm{x}}^{2}+\dots +{\mathrm{C}}_{\mathrm{n}}{\mathrm{x}}^{\mathrm{n}}-----\left(\mathrm{i}\right)$

on multiplying Eqs. (i) and (ii) and taking the
coefficient of constant terms in right hand side

$={\mathrm{C}}_{0}^{2}+{\mathrm{C}}_{1}^{2}+{\mathrm{C}}_{2}^{2}+\dots +{\mathrm{C}}_{\mathrm{n}}^{2}$

In right hand side

term containing xn in (1 + x)2n.

clearly, the coefficient of xn in (1 + x)2n is equal to

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