a)
$\begin{array}{l}
C_n^k + 3C_n^{k - 1} + 3C_n^{k - 2} + C_n^{k - 3}\\
= \left( {C_n^k + C_n^{k - 1}} \right) + 2\left( {C_n^{k - 1} + C_n^{k - 2}} \right) + \left( {C_n^{k - 2} + C_n^{k - 3}} \right)\\
= C_{n + 1}^k + 2C_{n + 1}^{k - 1} + C_{n + 1}^{k - 2}\\
= \left( {C_{n + 1}^k + C_{n + 1}^{k - 1}} \right) + \left( {C_{n + 1}^{k - 1} + C_{n + 1}^{k - 2}} \right)\\
= C_{n + 2}^k + C_{n + 2}^{k - 1} = C_{n + 3}^k
\end{array}$
b)
$\begin{array}{l}
{\left( {1 + x} \right)^n} = C_n^0 + C_n^1x + C_n^2{x^2} + ... + C_n^n{x^n}\\
{\left( {x + 1} \right)^n} = C_n^0{x^n} + C_n^1{x^{n - 1}} + C_n^2{x^{n - 2}} + ... + C_n^n\\
\Rightarrow {\left( {1 + x} \right)^n}{\left( {x + 1} \right)^n}\\
= \left( {C_n^0 + C_n^1x + C_n^2{x^2} + ... + C_n^n{x^n}} \right)\left( {C_n^0{x^n} + C_n^1{x^{n - 1}} + C_n^2{x^{n - 2}} + ... + C_n^n} \right)\\
\Leftrightarrow {\left( {1 + x} \right)^{2n}} = \left( {C_n^0 + C_n^1x + C_n^2{x^2} + ... + C_n^n{x^n}} \right)\left( {C_n^0{x^n} + C_n^1{x^{n - 1}} + C_n^2{x^{n - 2}} + ... + C_n^n} \right)\\
Ta\,co:\\
{\left( {1 + x} \right)^{2n}} = C_{2n}^0 + C_{2n}^1x + ... + C_{2n}^n{x^n} + ... + C_{2n}^{2n}{x^{2n}}\\
\Rightarrow He\,so\,cua\,{x^n}\,la\,C_{2n}^n\\
Lai\,co:\\
\left( {C_n^0 + C_n^1x + C_n^2{x^2} + ... + C_n^n{x^n}} \right)\left( {C_n^0{x^n} + C_n^1{x^{n - 1}} + C_n^2{x^{n - 2}} + ... + C_n^n} \right)\\
\Rightarrow He\,so\,cua\,{x^n}\,la:\\
{\left( {C_n^0} \right)^2} + {\left( {C_n^1} \right)^2} + ... + {\left( {C_n^n} \right)^2}\\
\Rightarrow C_{2n}^n = {\left( {C_n^0} \right)^2} + {\left( {C_n^1} \right)^2} + ... + {\left( {C_n^n} \right)^2}
\end{array}$
