The pigeonhole principle states that if more objects are placed into fewer containers, at least one container must receive more than one object.

If $n+1$ objects are assigned to $n$ boxes, some box contains at least two objects.

More generally, if $N$ objects are distributed among $k$ boxes, some box contains at least

$$\left\lceil\frac Nk\right\rceil$$

objects.

The power of the principle is choosing what counts as an object and what counts as a box. A complicated existence claim can become a simple comparison of finite counts.

Pigeonhole arguments prove that a repeated value, collision or sufficiently crowded class must exist without needing to identify it in advance.