Difference between revisions of "Power's of 2 in pascal's triangle"
| Line 39: | Line 39: | ||
=== Theorem === | === Theorem === | ||
| − | It states that <math>\binom{n}{0}+\binom{n}{1}+...+{n}{n}</math>. | + | It states that <math>\binom{n}{0}+\binom{n}{1}+...+\binom{n}{n}</math>. |
=== Why do we need it? === | === Why do we need it? === | ||
Revision as of 16:37, 16 June 2019
Contents
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
Combanations
Pascal's Triangle can also be written like this
And on and on...
Remember that 
where 
.
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 
.
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
Subset proof
Say you have a word with n letters. How many subsets does it have in terms of n? Here is how you answer it: You ask the first letter Are you in or are you out? Same to the second letter. Same to the third. Same to the n. Each of the letters has two choices: In and Out. The would be 
2^n$.
Alternate proof
If you look at the way we built the triangle you see that each number is row n-1 is added on twice in row n. This means that each row doubles. That means you get powers of two.
