Motivating combinatorics identity to different audiences: $ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $

Question

Recently I asked the following identity on Math.StackExchange knowing it had several proofs:

$$ \binom{n}{k}^2 \geq \binom{n}{k-1}\binom{n}{k+1} $$

See here. They quickly give the one-liner:

In your initial inequality divide the left hand side by the right hand side and simplify.

This is kind of a proof. It works on math.SE since many people know combinatorics there.
Probably they meant:

Using the identity $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ and cross multiplying, the identity is equivalent to: $$ k!^2 (n-k)!^2 \leq (k-1)!(k+1)! (n-k-1)! (n-k+1)! $$

Let's try proving: $k!^2 \geq (k-1)!(k+1)!$ Dividing both sides by $k!^2$ we get $1 \leq \frac{k+1}{k-1}$ which is true.

A similar proof works for the factors of $(n-k)!$

Overall, people had no idea where I was coming from. I had been reading a Richard Stanley article about log-concave sequence of numbers and I was hoping to digest it for my class. The logic made sense to me:

• Instead of clearing denominators every single time (which maybe neither exciting nor informative), why not exploit the that $a_n^2 \geq a_{n-1}a_{n+1}$ for this particular sequence of numbers?

Eventually two other proofs surfaced using other branches of math, but I still don't feel like I got any point across.

In fact, how do I motivate even this one proof? Is this equation really that obvious that it only deserves one sentence?

Answer 1

Does this help?

$(k-1)!(k+1)! = (\frac{k!}{k})(k!(k+1)) = k!^2\frac{k+1}{k} \ge k!^2\frac{k}{k}=k!^2$

and for $(n-k)!$ define $a=(n-k)$ and use the same trick.

Another way would be to note that ${n\choose{k-1}} = \frac{1}{k}{n\choose{k}}$ and that ${n\choose{k+1}} = k{n\choose{k}}$

Answer 2

It depends on the audience... I (and you, clearly) don't need more; my second year students (struggling with basic factorial identities and such) need a lot more handholding.

Question then is how to do the handholding. I have found that it is helpful to state what proof technique will be used (even if "obvious" from the proof itself), and write out most steps, even if rather simple algebra. If on the blackboard, you can skimp steps, but be prepared to fill in when asked (and you end up spending more time that way). This helps when fine-tuning.