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  4. The value of the sum (nC1)2+(nC2)2+(nC3)2+ ... +(nCn)2 is

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Q. The value of the sum $\left(^{n}C_{1}\right)^{2}+\left(^{n}C_{2}\right)^{2}+\left(^{n}C_{3}\right)^{2}+ ... +\left(^{n}C_{n}\right)^{2}$ is

WBJEEWBJEE 2014Binomial Theorem

Solution:

We know that
$(1+x)^{n}={ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2} +\ldots+{ }^{n} C_{n} x^{n}\,\,\,\,...(i)$
and $(x+1)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} +{ }^{n} C_{2} x^{n-2}+\ldots+{ }^{n} C_{n}\,\,\,\,...(ii)$
On multiplying Eqs. (i) and (ii), we get
$(1+x)^{2 n}=\left({ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\ldots+{ }^{n} C_{n} x^{n}\right)$
$\times\left({ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1}+{ }^{n} C_{2} x^{n-2}+\ldots+{ }^{n} C_{n}\right)$
Coefficient of $x^{n}$ in RHS
$=\left({ }^{n} C_{0}\right)^{2}+\left({ }^{n} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}$
and coefficient of $x^{n}$ in LHS $={ }^{2 n} C_{n}$
$\therefore \left({ }^{n} C_{0}\right)^{2}+\left({ }^{n} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}=\frac{2 n !}{n ! n !} $
$\Rightarrow \left({ }^{n} C_{1}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}=\frac{(2 n) !}{n ! n !}-1$
$={ }^{2 n} C_{n}-1$