An interesting theorem related to Pascal’s triangle is the hockey stick theorem or the christmas stocking theorem. This theorem states that the sum of a diagonal in Pascal’s triangle is the element perpendicularly under it (image from Wolfram Mathworld):
So here the sum of the blue numbers , 1+4+10+20+35+56, is equal to the red number, 126. It is easy to see why it’s called the hockey stick theorem – the shape looks like a hockey stick!
An alternative, algebraic formulation of the hockey stick theorem is follows:
But this works in two ways, considering the symmetry of Pascal’s triangle. The flipped version would be:
Using Pascal’s identity, it is fairly trivial to prove either identity by induction. Instead I present an intuitive, animated version of the proof:
Lucky's Notes 
I think the first summation should be n+i choose i rather than n+1 choose i (and second should change accordingly as well)
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