Power's of 2 in pascal's triangle

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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:

      1
     1 1
    1 2 1
   1 3 3 1
  1 4 6 4 1

And on and on...

Patterns and properties

Conbanations

Pascal's Triangle can also be written like this

                           $\binom{0}{0}$
                $\binom{1}{0}$                  $\binom{1}{1}$
   $\binom{2}{0}$                     $\binom{2}{1}$                $\binom{2}{1}$

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

Sum of rows

                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).

Powers of two

Theorem

Theorem

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

Why do we need it?

It is useful is many word problems (That means, yes, you can use it in real life) and it is just a cool thing to know. More at https://artofproblemsolving.com/videos/mathcounts/mc2010/419.

Proof

Long proof

Short proof

Which proof do you like better?