<s>[INST] What are some easy but beautiful patterns in Pascal's Triangle? Pascal's triangle has wide applications in Mathematics.I have seen that the most important applications relate to the binomial coefficients and combinatorics.
Are there some other beautiful interesting patterns in Pascal's triangle that can be found by selecting some other combination modes ?
 [/INST] 
There is a nice hexagonal property in   the  Pascal   triangle:
\begin{array}{ccccccccccccccccc}
&&&&&&&&1\\
&&&&&&&1&&1\\
&&&&&&1&&2&&1\\
&&&&&1&&3&&3&&1\\
&&&&1&&4&&6&&4&&1\\
&&&1&&\mathbf{\color{blue}{5}}&&\mathbf{\color{blue}{10}}&&10&&5&&1\\
&&1&&\mathbf{\color{blue}{6}}&&\mathbf{\color{red}{15}}&&\mathbf{\color{blue}{20}}&&15&&6&&1\\
&1&&7&&\mathbf{\color{blue}{21}}&&\mathbf{\color{blue}{35}}&&35&&21&&7&&1\\
1&&8&&28&&56&&70&&56&&21&&8&&1\\
\end{array}
  The hexagon around $\binom{n}{k}$ fulfils
  \begin{align*}
\binom{n-1}{k-1}\binom{n}{k+1}\binom{n+1}{k}&=\binom{n-1}{k}\binom{n}{k-1}\binom{n+1}{k+1}\\
\end{align*}

The  example  $\binom{n}{k}=\binom{6}{2}=\color{red}{15}$ shows
\begin{align*}
\binom{n-1}{k-1}\binom{n}{k+1}\binom{n+1}{k}&=\binom{5}{1}\binom{6}{3}\binom{7}{2}=\color{blue}{5}\cdot \color{blue}{20}\cdot   \color{blue}{21}=2100\\
\binom{n-1}{k}\binom{n}{k-1}\binom{n+1}{k+1}&=\binom{5}{2}\binom{6}{1}\binom{7}{3}=\color{blue}{10}\cdot \color{blue}{6}\cdot \color{blue}{35}=2100
\end{align*}

In  other words:
  
  
*
  
*A hexagon around a binomial coefficient $\binom{n}{k}$ is always a perfect square. :-)
  

See Generalized hidden hexagon squares by A.K. Gupta for a generalized version of this relationship.
</s>