Proving the Evenness of (\stackrel{2n}{n}) Using the Binomial Theorem

[ipitydatfu]

Homework Statement

prove that ($$\stackrel{2n}{n}$$) is even when n $$\geq1$$

Homework Equations

as a hint they gave me this identity:
$$\stackrel{n}{k}$$= (n/k)($$\stackrel{n-1}{k-1}$$)

The Attempt at a Solution

by using that identity i got:

($$\stackrel{2n}{n}$$) = (2n/n) ($$\stackrel{2n-1}{n-1}$$)
= (2) ($$\stackrel{2n-1}{n-1}$$)

i thought anything multiplied by 2 is an even number. but then again this is discrete math. how would i inductively show that this is true?

 

[HallsofIvy]

That's pretty much it. The definition of "even number" is that it is of the form 2k for some integer k. Do you already know that $\left(\begin{array}{c}n\\2i\end{array}\right)$ is always an integer?

 

[ipitydatfu]

oh yeah! I forgot about that! thanks!