Difference between revisions of "Power's of 2 in pascal's triangle"
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= Powers of two = | = Powers of two = | ||
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| + | == Theorem == | ||
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| + | === Theorem === | ||
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| + | The theorem is this: <math>\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}</math>. | ||
| + | |||
| + | === 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. | ||
| + | |||
| + | == Proof == | ||
Revision as of 15:11, 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
Conbanations
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
The theorem is this: 
.
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.
