Present Exchange

An office of twenty people decides to engage in a holiday ‘Secret Santa’ assignment. Each staff member purchases a dozen small gifts. Each employee distributes their gifts to other members of the office, making sure each recipient, at most, receives one gift from them (meaning staffers get one, or no gifts, from any individual giver; nobody gets more than one gift from any single person). Also, people do not gift to themselves!

Prove that there must be occurrences where a pair of employees mutually exchange gifts with each other.