We need to bound the binomial coefficients a lot of times. In this post, we will prove bounds on the coefficients of the form and where and is an integer.

Proposition 1.For positive integers such that ,

Proposition 2.For a positive integer and any such that and ,

where the binary entropy function is defined as follows:

Proposition 3.For a positive integer and any such that and ,where is the binary entropy function.

**Proof of Proposition 1**

First, recall the power series . For , this sum is definitely larger than its th term only. In other words, , which implies . Now,

This proof is extracted from the StackExchange discussion here. For more inequalities/bounds for binomial coefficients, see Wikipedia.

**Proof of Proposition 2**

Let us briefly recall Stirling’s approximation of .

Theorem(Stirling’s approximation). For any positive integer ,where

**Proof of** Proposition** 3**

First, note that implies .

We remark that the last approximation is rather loose.