The distance between the points $(1, 2)$ and $(4, 6)$ is $5$ units.

To calculate the distance between two points on a coordinate plane, we utilize the distance formula, which is derived from the Pythagorean theorem. The distance formula is expressed as:

In this formula, $(x_1, y_1)$ and $(x_2, y_2)$ represent the coordinates of the two points. For our specific points, we assign $(x_1, y_1) = (1, 2)$ and $(x_2, y_2) = (4, 6)$.

First, we calculate the differences in the x-coordinates and y-coordinates as follows:

Next, we square these differences:

We then sum these squared differences:

Finally, we take the square root of this result to determine the distance:

Thus, the distance between the points $(1, 2)$ and $(4, 6)$ is indeed $5$ units. This method can be applied to any pair of points on a coordinate plane, making it an invaluable tool in geometry.