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...

Patterns and properties

Conbanations

Pascal's Triangle can also be written like this

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

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

The theorem is this: $\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$ .

Why do we need it?

You would need for counting the number of subsets in a word, The number of ways people could volunteer for something and many other things. It's also a cool thing to know about that your friends don't.

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Power's of 2 in pascal's triangle

Contents

Review

Pascal's Triangle

Patterns and properties

Conbanations

Sum of rows

Powers of two

Theorem

Theorem

Why do we need it?

Proof