Definition

Definition. The degree of freedom of a system is the minimum number of independent coordinates needed to completely specify the position and configuration of that system in space.

Examples: (1) a free particle constrained to move only along the x-axis needs just one coordinate ($x$) to fix its position, so it has one degree of freedom. (2) A particle free to move anywhere over a flat plane needs two coordinates ($x,y$), so it has two degrees of freedom. (3) A particle free to move anywhere in three-dimensional space needs three coordinates ($x,y,z$), so it has three degrees of freedom.

General counting rule. For a sample of $N$ gas molecules, each treated as a free point particle in space, the total number of degrees of freedom of the whole system would be $f=3N$ (three per molecule). But if the system is subject to $q$ independent constraints -- restrictions that reduce its freedom to move, such as atoms within a molecule being held at a fixed separation from each other -- then the number of independent degrees of freedom is reduced accordingly:

$f = 3N - q.$ …