Let n be an element of the positive numbers (Z+). Prove that 3 divides (3n n) or "3n choose n". Use the definition of a binomial coefficient to solve.
Definition of a Binomial Coefficient: (n k) := ( n! / k!(n - k)! )
The Attempt at a Solution
I've done the basics. I've replaced n and k with 3n and n, making the equation: (3n n) = ( 3n! / n!(3n - n)! ), then simplifying to ( 3n! / n!(2n)! ), which equals just ( 3 / 2n! ).
If it is divisible by 3, I suppose this can be expressed as: (( 3 / 2n! )) / 3 = k, and therefore, ( 3 / 2n! ) = 3k . It seems like the real proof here is in showing that 2n! is an integer.
How do I go forward?