Pascal´s triangle

Do a mathematical exploration on pascals triangle. This comes under the topic binomial theorem.
But the MAIN focus should be on Pascal´s triangle.
The file attached includes 2 examples on different topic
plus the other two pictures is a brief info on my topic selected.
you may come up with new solving pascals ideas self or online sources to help.
I request that sources used should be easily available on the net (not concerned from books)
The task should not only include theories but solving problems too