Question

Why does pascal's triangle give the coefficients to binomials? The pattern is useful and handy, but why does it work, what's the relation there?

Answer 1

Works due to this property
\[ \binom{n}{r-1}+ \binom{n}{r} = \binom{n+1}{r}\]
You put 1 's at both sides which are binomial coefficient's and rest come this way.

Answer 2

Wikipedia has a couple of proofs one suggested under Pascal's Triangle and a few under Binomial Theorem. The proof is fairly straight forward if you are used to proofs. If you read them and understand them, great, otherwise you might want to have someone explain it step by step or, if necessary, keep learning until you have the math background to follow the proof(s). I used to know all that, but I would need to do a bit (quite a bit) of brushing up to try and explain it.

Answer 3

http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/PascalTriangleAnimated2.gif/220px-PascalTriangleAnimated2.gif

Answer 4

I don't think this explained it to me very well. Anyone else willing to try to explain the proof?

Answer 5

what do you think binomial coefficients are?

Answer 6

Constants that add up from multiplication and having repeating terms. I suppose the connection doesn't seem very obvious to me.