Thursday, May 14, 2009

BoB oh BoB

Hello. This last unit was called combinatorics and I found it very interesting because there are so many different ways and different ways of thinking that we can solve problems with. This is the kind of math we could use in every day situations. First things first, definitions:

Combination: In combinatorial mathematics, a combination is an un-ordered collection of distinct elements, usually of a prescribed size and taken from a given set. \mathbf{C}(n,k) = \mathbf{C}_k^n= {_nC_k} = {n \choose k} = \frac{n!}{k!(n-k)!}.
Permutation:In combinatorics, a permutation is usually understood to be a sequence containing each element from a finite set once, and only once
 P(n, r) = \frac{n!}{(n-r)!}.
(Wikipedia definitions)

It is very crucial to understand these terms.

For me, everything in this unit went swimmingly well until we learned about this:
(x + y)^2 = x^2 + 2xy + y^2\,
(x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3\,
(x + y)^4 = x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4\,
(x + y)^5 = x^5 + 5x^4y + 10x^3y^2 + 10x^2y^3 +5xy^4 + y^5.\,

(x+y)^n=\sum_{k=0}^n{n \choose k}x^{n-k}y^{k}\quad\quad\quad(1)

oh boy... But I realized after we did some questions, and with further exploring this, everything went a whole lot better...

File:Pascal's triangle 5.svg
That is pascals triangle. Don't forget some nifty tricks in the triangle, like 2^x rule and how it corresponds with the row number, and the 11^x, and also the good old hockey stick pattern.

To view all possible poker hand probabilities, go to this link!

I didn't find much trouble with the poker hands because I'm a big fan of the game.

Well this sums up my bob post, the next bob is bob by the way!


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