If n objects are arranged in a row, then the number of ways of selecting three of these objects so that no two of them are next to each other, is

Let x be the number of objects to the left of the first object chosen, y the number of objects between the first and the second, z the number of objects between the second and the third and u the number of objects to the right of the third object. Then,

$x,u\ge 0;y,z\ge 1$ and $x+y+z+u=n-3$

Let $y_1=y-1$ and $z_1=z-1$. Then, $y_1\ge 0,z_1\ge 0$ such that $x+y_1+z_1+u=n-5$.

The total number of non-negative integral solutions of this equation is 

${ }^{n-5+4-1}{C}_{4-1}{=}^{n-2}{C}_3$.