Pascal’s triangle has long and celebrated history, see this TedEd video:

What makes it more interesting is its relations with various domains of Mathematics (if you don’t understand the three relations discussed, refer Wikipedia). Here I will point few such connections:

1. **Fibonacci Numbers**: I believe that this is the most celebrated observation from pascal’s triangle. To see the jungle of Equations involved, visit http://www.maplesoft.com/applications/view.aspx?SID=3617&view=html

2. **Chebyshev Polynomial:** We can find coefficients of Chebyshev Polynomial using pascal’s triangle. See- http://mathpages.com/home/kmath304.htm

3.** Ramsey Number**: Upper bound of Ramsey Number can be found using Pascal’s triangle, for more details refer : https://plus.maths.org/content/friends-and-strangers