Often a particular probability problem is not binomial at all. This happens when there is no repetition of equivalent experiments and/or there are more than two possible outcomes ("success" or "failure").

For example, imagine you have a small population of, say, $25$ elements, $5$ of which have a particular characteristic. From this population you take a sample of $6$ elements. $X$ would then be the number of elements in your sample that have the specific characteristic. The corresponding probabilities are:

${\rm P}(X=2)=\frac{5}{25}\cdot \frac{4}{24}\cdot \frac{15}{23}\cdot \frac{14}{22}\cdot \frac{13}{21}\cdot \frac{12}{20}\cdot \left(\begin{array}{c}6\\ 2\end{array}\right)$.

The expected value is: ${\rm E}\left(X\right)=25\cdot \frac{5}{25}$.

If you take a small sample from a large population (for example $6$ out of $25000$, of which $5000$ have a particular characteristic) you can still use the binomial probability model, even though you are not actually dealing with equal chances. This is because the value of the fractions $\frac{5000}{25000}$ and $\frac{4999}{24999}$ is almost the same.
In practice, when you take a sample from a large population, and you are interested in the number of elements that carry a particular characteristic, then you can simply use the binomial probability model.