binomial coefficient (3n chose n) is divisible by 3

[voyage200]

I have to prove that the binomial coefficint (3n) is divisible by 3 for all positive
__________________________________( n )
integers n.

where:
(3n) = 3n!/(n! (3n-n!)=3/(2n)!
(n )
but I don't know how to prove its divisible by 3. Please help.

 

[marcmtlca]

\(\displaystyle \Large {3n \choose n}\) is divisible by 3?

\(\displaystyle =\Large \frac{(3n)!}{(3n-n)!n!}=\frac{(3n)!}{(2n)!(n)!}=\frac{(3n)(3n-1)!}{(3n-1-(n-1) )!(n-1)!(n)}=\frac{(3n)}{(n)} {3n-1 \choose n-1}=3 {3n-1 \choose n-1}\)

All that is left is to notice that \(\displaystyle {3n-1 \choose n-1}\) is always an integer.