Proving two equations using multinomial coefficients.

Question

Use the fact that multinomial coefficients are non-negative integers to prove that (a) $(2n)!$ is divisible by $2^n$ (and that for n >= 2 the quotient is even) and (b) $(n^2!)!$ is divisible by $(n!)^{n+1}$

By "divisible by" the problem means evenly divide.

I have done similar problems to these using regular induction (although they did not have a variable in the denominator). I am not sure how to do these using multinomial coefficients. How do I start these two problems?

Answer 1

HINT:

• What is $$\binom{2n}{\underbrace{2,\ldots,2}_n}$$ when expressed in terms of factorials? For the second part of (a) try to cancel all of the factors in the denominator.

• Is there an integer $m$ that makes $$\binom{n^2!}{\underbrace{n!,\ldots,n!}_{n+1},m}$$ a multinomial coefficient? What is that coefficient when expressed in terms of factorials?