Entropy and multinomial coefficients

Multinomial coefficients have exponential growth rates governed by entropy. This connection underlies many results in information theory and large deviations, including the method-of-types.

Multinomial coefficient and empirical proportions

Let $n_1,\dots,n_k$ be nonnegative integers with $\sum_{i=1}^k n_i = n$ . The multinomial coefficient is

{n \choose n_1,\dots,n_k} \;=\; \frac{n!}{\prod_{i=1}^k n_i!},

where $n! = n(n-1)\cdots 1$ is the factorial.

Define proportions $p_i = n_i/n$ (so $\sum_i p_i = 1$ ). The (natural-log) Shannon entropy is

H(p) = -\sum_{i=1}^k p_i \log p_i.

(Using base-2 logarithms multiplies all entropy expressions by $\log 2$ .)

Main asymptotic principle

Using stirlings-approximation, one obtains

\log {n \choose n_1,\dots,n_k} = n H(p) + O(\log n) \quad (n\to\infty),

as long as the number of categories $k$ is fixed (and typically assuming each $p_i$ is bounded away from $0$ to avoid singular prefactors).

A more detailed approximation (when all $p_i>0$ ) is

{n \choose n_1,\dots,n_k} \approx \frac{e^{nH(p)}}{(2\pi n)^{(k-1)/2}\sqrt{p_1p_2\cdots p_k}},

where “ $\approx$ ” means the ratio tends to $1$ under standard regularity conditions.

Simple entropy bounds (polynomial accuracy)

There are widely used bounds that capture the correct exponential rate without tracking constants:

(n+1)^{-k}\,e^{nH(p)} \;\le\; {n \choose n_1,\dots,n_k} \;\le\; e^{nH(p)}.

These bounds are especially convenient when only exponential rates matter (e.g., in large deviation estimates).

Proof idea for the upper bound

Apply the multinomial theorem with weights $p_i$ :

1 = \Bigl(\sum_{i=1}^k p_i\Bigr)^n = \sum_{\sum n_i=n} {n \choose n_1,\dots,n_k}\prod_{i=1}^k p_i^{n_i}.

Every term is nonnegative, so for the particular $(n_1,\dots,n_k)$ ,

{n \choose n_1,\dots,n_k}\prod_{i=1}^k p_i^{n_i} \le 1,

hence

{n \choose n_1,\dots,n_k} \le \prod_{i=1}^k p_i^{-n_i} = e^{nH(p)}.

Why this matters

The exponential size of a “type class” (all sequences with the same empirical proportions) is controlled by $e^{nH(p)}$ , up to polynomial factors; this is the core counting step in the method-of-types.
Many probability bounds reduce to comparing $e^{nH(p)}$ with likelihood terms, producing Kullback–Leibler divergence rates.