Problem

Source: China south east mathematical olympiad 2006 day1 problem 3

Tags: algorithm, geometry, geometric transformation, combinatorics unsolved, combinatorics



There is a standard deck of $52$ cards without jokers. The deck consists of four suits(diamond, club, heart, spade) which include thirteen cards in each. For each suit, all thirteen cards are ranked from “$2$” to “$A$” (i.e. $2, 3,\ldots , Q, K, A$). A pair of cards is called a “straight flush” if these two cards belong to the same suit and their ranks are adjacent. Additionally, "$A$" and "$2$" are considered to be adjacent (i.e. "A" is also considered as "$1$"). For example, spade $A$ and spade $2$ form a “straight flush”; diamond $10$ and diamond $Q$ are not a “straight flush” pair. Determine how many ways of picking thirteen cards out of the deck such that all ranks are included but no “straight flush” exists in them.