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I've seen this a couple times. What does the part where y is over k with a space between them? What does this imply? (Below is the Binomial Theorem equation)

$$(x + a)^y = \sum\limits_{k=1}^{\infty}\binom{y}{k}x^ka^{y-k}$$

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marked as duplicate by Daniel Fischer Feb 16 '15 at 22:37

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up vote 8 down vote accepted

It is a binomial coefficient. The symbol is typically defined by $$\binom{n}{k}=\frac{n!}{(n-k)!\times k!}$$ where $n$ and $k$ are non-negative integers, and the exclamation point $!$ denotes the factorial.

However, in the example you cite, which is often called the generalized binomial theorem, in place of the integer $n$ we can actually use any real number $\nu$ (this is the Greek letter nu), and we define $$\binom{\nu}{k}=\frac{\nu\times(\nu-1)\times\cdots\times(\nu-k+1)}{k!}$$ This agrees with the standard definition when $\nu$ is a non-negative integer.

For an example of how this symbol is computed, $$\binom{5}{2}=\frac{5!}{3!\times 2!}=\frac{5\times 4\times 3\times 2\times 1}{(3\times 2\times 1)\times(2\times 1)}=\frac{120}{12}=10$$

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Many texts in probability theory sometimes write the binomial coefficient $\binom{n}{k}$ as ${}_nC_k$, where it is called "$n$ choose $k$."

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Usually it's conventional to have the $C$ large and the $n$ and $k$ small -- that is, ${}_n C_k$. (I'm not sure I'd think of this as the usual notation; the ${n \choose k}$ notation seems more common among "grown up" mathematicians, although from talking to my students I think the $C$ notation is still taught at the high school level.) – Michael Lugo Feb 5 '12 at 23:31

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