Problem:
Robert has indistinguishable gold coins and indistinguishable silver coins. Each coin has an engraving of a face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the coins.
Solution:
First consider the orientation of the coins. Label each coin or depending upon whether it is face up or face down, respectively. Then for each arrangement of the coins, there is a corresponding string consisting of a total of eight U's and D's that is formed by listing each coin's label starting from the bottom of the stack. An arrangement in which no two adjacent coins are face to face corresponds to such a string that does not contain . Thus the first in the string must have no 's after it. The first may appear in any of eight positions or not at all, for a total of nine allowable strings. For each of these nine strings, there are ways to pick the positions for the four gold coins, and the positions of the silver coins are then determined. Thus there are arrangements that satisfy Robert's rules of order.
The problems on this page are the property of the MAA's American Mathematics Competitions