$\begin{array}{l}
1)C_n^k + 3C_n^{k - 1} + 3C_n^{k - 2} + C_n^{k - 3}\\
= \left( {C_n^k + C_n^{k - 1}} \right) + 2\left( {C_n^{k - 1} + C_n^{k - 2}} \right) + \left( {C_n^{k - 2} + C_n^{k - 3}} \right)\\
= C_{n + 1}^k + 2C_{n + 1}^{k - 1} + C_{n + 1}^{k - 2}\\
= \left( {C_{n + 1}^k + C_{n + 1}^{k - 1}} \right) + \left( {C_{n + 1}^{k - 1} + C_{n + 1}^{k - 2}} \right)\\
= C_{n + 2}^k + C_{n + 2}^{k - 1} = C_{n + 3}^k
\end{array}$
