A simple inequality

[santi_spadaro at virgilio.it]

There's an old Monthly article (Vol. 87, n.9) of Solomon Golomb which deals
with iterated binomial coefficients. Among the many elegant results which
are proved I came to this simple inequality:

Binomial (Binomial (n,2), 3) > Binomial (Binomial (n,3), 2)

While providing both an algebraic and combinatorial proof for most of the
other identities, here he only gives a proof based on numerical estimates
of the Binomial coefficients. This result also follows trivially from two
of the reduction formulas which Golomb states at the end of the article,
namely:

Binomial (Binomial (n,2),3)= Binomial (n+1,6)+13*Binomial (n+2, 6)+Binomial
(n+3,6)

Binomial (Binomial (n,3),2) = 6*Binomial (n+2,6)+3*Binomial (n+1,6)+Binomial
(n,6)

I would be very pleased to see a combinatorial proof of the above fact,
say an injection between 3-multigraphs on 2 edges and n vertices and graphs
on 3 edges and n vertices. Can you think of any?

Regards,
Santino