1. Tardigrade
  2. Question
  3. Mathematics
  4. The value of the sum ( nC1)2+( nC2)2+( nC3)2+...+( nCn)2 is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

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

Binomial Theorem

Solution:

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}
$