Question

The value of the sum $(\,^nC_1)^2+(\,^nC_2)^2+(\,^nC_3)^2+...+(\,^nC_n)^2$ is

• A. $(\,^{2n}C_n)^2$
• B. $\,^{2n}C_n$
• C. $\,^{2n}C_n+1$
• D. $\,^{2n}C_n-1$

Answer 1

The Correct Option is D

We know that $ (1+ x )^{ n }={ }^{ n } C _{0}+{ }^{ n } C _{1} x +{ }^{ n } C _{2} x ^{2}+\cdots+{ }^{ n } C _{ n } x ^{ n } \quad \ldots \ldots \text { (i) } $ and $ (x+1)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1}+{ }^{n} C_{2} x^{n-2}+\cdots+{ }^{n} C_{n} $ On multiplying equations (i) and (ii), we get $ \begin{array}{l} (1+x)^{2 n}=\left({ }^{n} C_{0}+{ }^{n} C_{1} x+{ }^{n} C_{2} x^{2}+\cdots+{ }^{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}+\cdots+{ }^{n} C_{n}\right) \end{array} $ Coefficient of $x ^{ n }$ in right hand side $=\left({ }^{ n } C _{0}\right)^{2}+\left({ }^{ n } C _{1}\right)^{2}+\cdots+\left({ }^{ n } C _{ n }\right)^{2}$ and $ \begin{array}{l} \text { coefficient of } x ^{ n } \text { in left hand side }={ }^{2 n } C _{ n } \\ \therefore\left({ }^{ n } C _{0}\right)^{2}+\left({ }^{ n } C _{1}\right)^{2}+\cdots+\left({ }^{ n } C _{ n }\right)^{2}=\frac{2 n !}{ n ! n !} \\ \Rightarrow\left({ }^{ n } C _{1}\right)^{2}+\cdots+\left({ }^{ n } C _{ n }\right)^{2}=\frac{(2 n ) !}{ n ! n !}-1={ }^{2 n } C _{ n }-1 \end{array} $