**Bounds and Asymptotic Formulas**

The following bounds for hold:

Stirling's approximation yields the bounds:

- and, in general, for
*m*≥ 2 and*n*≥ 1,

and the approximation

- as

The infinite product formula (cf. Gamma function, alternative definition)

yields the asymptotic formulas

as .

This asymptotic behaviour is contained in the approximation

as well. (Here is the *k*-th harmonic number and is the Euler–Mascheroni constant).

The sum of binomial coefficients can be bounded by a term exponential in and the binary entropy of the largest that occurs. More precisely, for and, it holds

where is the binary entropy of .

A simple and rough upper bound for the sum of binomial coefficients is given by the formula below (not difficult to prove)

Read more about this topic: Binomial Coefficient

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