# Binomial Coefficients

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• February 1st 2010, 02:23 PM
adkinsjr
Binomial Coefficients
Binomial coefficients are defined:

$\left(\begin{array}{c}n\\k\end{array}\right)=\frac {n!}{k!(n-k)!}$

I'm not sure how two write these out going from $k={0,1,2,...,n}$. For $k=0$ I have:

$\left(\begin{array}{c}n\\0\end{array}\right)=\frac {n!}{!(n-0)!}=\frac{n!}{1!n!}=\frac{n!}{n!}=1$

For $k=1$

$\left(\begin{array}{c}n\\1\end{array}\right)=\frac {n!}{1!(n-1)!}=\frac{n!}{(n-1)!}$

I know this should equal $n$, but I don't know how to show that.
• February 1st 2010, 02:25 PM
Drexel28
Quote:

Originally Posted by adkinsjr
Binomial coefficients are defined:

$\left(\begin{array}{c}n\\k\end{array}\right)=\frac {n!}{k!(n-k)!}$

I'm not sure how two write these out going from $k={0,1,2,...,n}$. For $k=0$ I have:

$\left(\begin{array}{c}n\\0\end{array}\right)=\frac {n!}{!(n-0)!}=\frac{n!}{1!n!}=\frac{n!}{n!}=1$

For $k=1$

$\left(\begin{array}{c}n\\1\end{array}\right)=\frac {n!}{1!(n-1)!}=\frac{n!}{(n-1)!}$

I know this should equal $n$, but I don't know how to show that.

For the love's sake, $n!=n\cdot(n-1)\cdots=n\cdot (n-1)!$.
• February 1st 2010, 06:24 PM
adkinsjr
Ok, I wasn't aware of the formal definition of a factorial:

$n!=\left\{\begin{array}{cc}1,&\mbox{ if }
n=0\\n(n-1)!, & \mbox{ if } n>0\end{array}\right.$