Binomial Coefficients

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

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

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

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

For $\displaystyle k=1$

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

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

Binomial coefficients are defined:

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

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

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

For $\displaystyle k=1$

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

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

For the love's sake, $\displaystyle n!=n\cdot(n-1)\cdots=n\cdot (n-1)!$.
• Feb 1st 2010, 06:24 PM
$\displaystyle n!=\left\{\begin{array}{cc}1,&\mbox{ if } n=0\\n(n-1)!, & \mbox{ if } n>0\end{array}\right.$