Abstract
A poker hand consists of five cards which are drawn from a deck of 52 cards. There are 2, 598, 960 (=_??_) different poker hands. The total number of the hands which (i) contain two or more cards of the same rank, or (ii) consist of the same suit, or (iii) consist of circularly consecutive ranks ((ii) and (iii) are not necessarily exclusive) is 1, 299, 480=_??_/2. In this paper we consider a problem arising from this fact. Let us suppose that there is a deck of nk cards of n ranks and k suits, and a hand consists of r cards. Let S(n, k, r) be the total number of the hands which satisfy at least one of (i), (ii) and (iii). Is there any triplet (n, k, r) for r_??_3 other than (13, 4, 5) for which S(n, k, r)=_??_/2 holds. One partial answer is “No” if n_??_10, 000.
