Question 179
Let $n$ be a positive integer. Consider all arrangements of $n$ identical green coins, $n$ identical white coins and $n$ identical orange coins in a row. Each such arrangement of $3n$ coins can be considered as a sequence of *blocks*, where coins within a block have the same colour and any two adjacent blocks contain coins of two different colours. For example, for the case $n = 4$, the arrangement *GGGOOWOOGWWWW* is formed by 6 blocks, namely *GGG*, *OO*, *W*, *OO*, *G* and *WWW*.
Show that the average number of blocks, over all distinct arrangements of the $3n$ coins, can be expressed in the form $An+B$, and determine the values of the constants $A$ and $B$.
Show that the average number of blocks, over all distinct arrangements of the $3n$ coins, can be expressed in the form $An+B$, and determine the values of the constants $A$ and $B$.