I was working on a math problem that required me to figure out the general summation of $\binom{N}{0} + \binom{N}{1} + \ldots + \binom{N}{K}$.

I know that if $k = N$, the answer is $2^N$. But is there an answer for a general $k$?

For reference, this is the problem:

> 
> How many $N$-bit binary strings contain anywhere from none to $\frac{N}{2}$ zeros (inclusive)? $N$ is even.
> 
> 
> 

The answer involves recognizing that all possible combinations of strings that have zeroes can be represented by $\binom{N}{0} + \binom{N}{1} + \ldots + \binom{N}{N}$, and that representing the answer to the problem (anywhere from $0$ to $\frac{N}{2}$ zeroes) would be $\binom{N}{0} + \binom{N}{1} + \ldots + \binom{N}{N/2}$.

Hence my question.