Maximum of trinomial coefficient

Question

I read a proof about the simple random walk in 3 dimensions and couldn't understand the following statement: $$\frac{n!}{k!j!(n-k-j)!}$$ has the maximum when $\ k, j $ and $\ n-k-j$ are as close to $\frac{n}{3}$ as possible.

Does anyone show me a proof for this?

This may help. http://math.stackexchange.com/questions/1008249/maximizing-the-trinomial-coefficient — paw88789, Mar 29 '15 at 11:35
I assume the proof follows the same pattern as the one showing that the maximum binomial coefficient is the central binomial coefficient. — Lucian, Mar 29 '15 at 11:42
Thanks. I have read this argument, but I want to see a rigorous proof. I think of Stirling formula for each factorial term, but cannot derive the result from that. — user126028, Mar 30 '15 at 02:10

Salomo · Accepted Answer · 2015-03-29T14:40:13.580

1

That is equivalent to show how to minimize $a!b!c!$ with $a+b+c=n$. Suppose $a-b>1$, then replace $a,b$ by $a-1,b+1$ we will get a smaller product as $(a-1)b(b+1)<(a-1)ab$. By the symmetry of $a,b,c$, the result follows, since $a,b,c$ are as close to $n/3$ as possible means exactly each pair in $a,b,c$ has no differnce greater than $1$.

edited Mar 29 '15 at 14:40

answered Mar 29 '15 at 11:55

Salomo

2,843

Did you mean to minimize $a!b!c!$ with $a+b+c=n$? – hardmath Mar 29 '15 at 14:15
Thanks. I have read this argument, but I want to see a rigorous proof. I think of Stirling formula for each factorial term, but cannot derive the result from that. – user126028 Mar 30 '15 at 02:09
I think my answer is quite rigorous if you could have the ability to understand it... – Salomo Mar 30 '15 at 08:46

Maximum of trinomial coefficient

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