Analysis of pairs in a generalized deck of playing cards

In this study, we analyze the number of pairs (i.e., two cards of the same rank) in a set of cards randomly selected from a deck of playing cards. While the standard deck of playing cards comprises 52 cards excluding the joker with 4 suits and 13 ranks, our analysis considers a generalized deck where the numbers of suits and ranks can be set arbitrarily. We derive the exact formulas for the mean and variance of the number of pairs in a given number of cards randomly selected from this generalized deck, expressed using the Gauss hypergeometric function. Moreover, we derive the asymptotic behavior of the mean and variance as the number of ranks tends to infinity.