Review

Pascal's Triangle

Pascal's Triangle is a triangular array of numbers where each number is the sum of the two numbers above it. It Looks something like this:

And on and on...

Pascal's Triangle can also be written like this

And on and on... Remember that $\binom{n}{r}=\frac{n!}{k!(n-k)!}$ where $n \ge r$ .

                1     =1
               1+1    =2
              1+2+1   =4
             1+3+3+1  =8
            1+4+6+4+1 =16

These are powers of two. Let's prove it true. (Note: There are dozens of more patterns but it would have nothing to do with powers of two).

It states that $\binom{n}{0}+\binom{n}{1}+...+{n}{n}$ .