<s>[INST] The sum of $\binom{n}{0}\binom{3n}{2n}-\binom{n}{1}\binom{3n-3}{2n-3}+\binom{n}{2}\binom{3n-6}{2n-6}+\cdots \cdots $ 
The sum of $$\binom{n}{0}\binom{3n}{2n}-\binom{n}{1}\binom{3n-3}{2n-3}+\binom{n}{2}\binom{3n-6}{2n-6}-\cdots \cdots \cdots $$

$\bf{My\; Try::}$ We can write above sum  as $$\sum^{n}_{k=0}(-1)^k\binom{n}{k}\binom{3n-3k}{2n-3k} = \sum^{n}_{k=0}(-1)^k\binom{n}{k}\binom{3n-3k}{n}$$
So sum $$ = \sum^{n}_{k=0}(-1)^k\cdot \frac{n!}{k!\cdot (n-k)!}\times \frac{(3n-2k)!}{n!\cdot (2n-3k)!} = \sum^{n}_{k=0}(-1)^k \cdot \frac{(3n-2k)!}{k!\cdot (n-k)!\cdot (2n-3k)!}$$
Now i did not understand how can i solve it after that , help Required, Thanks
 [/INST] \begin{align*}
&\binom{n}{0}\binom{3n}{2n}-\binom{n}{1}\binom{3n-3}{2n-3}+\binom{n}{2}\binom{3n-6}{2n-6}-\cdots \\
&=\binom{n}{0}\binom{3n}{n}-\binom{n}{1}\binom{3n-3}{n}+\binom{n}{2}\binom{3n-6}{n}-\cdots
\end{align*}
The right hand side is the coefficient of $x^n$ in 
\begin{align*}
&\binom{n}{0}(1+x)^{3n} - \binom{n}{1}(1+x)^{3n-3} + \cdots \\
&= (1+x)^{3n}\left(1- \frac{1}{(1+x)^3}\right)^n \\
&= (1+x)^{3n-3}(3x+3x^2+x^3)^n \\
&= x^n(1+x)^{3n-3}(3+3x+x^2)^n
\end{align*}
and hence equals $3^n$.
</s>