Using Pinochle to motivate the restricted combinations with repetitions problem

Abstract

A standard example used in introductory combinatoric courses is to count the number of five-card poker hands possible from a straight deck of 52 distinct cards. A more interesting problem is to count the number of distinct hands possible from a Pinochle deck in which there are multiple, but obviously limited, copies of each type of card (two copies for single-deck, four for double deck). This problem is more interesting because our only concern is to count the number of distinguishable hands that can be dealt. In this note, under various scenarios, we will discuss two combinatoric techniques for counting these hands; namely, the inclusion–exclusion principle and generating functions. We will then show that these Pinochle examples motivate a general counting formula for what are called ‘regular’ combinations by Riordan. Finally, we prove the correctness of this formula using generating functions.