Binomial Coefficient and Factorial Notation

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Ok, so first thing - binomial coefficient. You know that it is...

n choose k = n!/[(k!(n-k)!].

this is basically a way to count, right. It is counting that is unordered without replacement. This means that order doesn't matter. {1,2,3} is the same thing as {2,1,3}. Without replacement means that for each observation you make, the sample size shrinks by one. For example if you are choosing lottery numbers, each time a number is selected, that number can no longer be selected.

There are other types of counting, too, right. Ordered, w/ and w/out replacement, and unordered with replacement, as well.

Ok, so the binomial coefficient has to do with the binomial theorem, which maybe you have seen.

(x+y)^n = sum(0,n)[n choose k] (x^k) (y^n-k)

where n choose k is the binomial coefficient.

Lets do an example...

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(x+y)^3 ...