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