In a jigsaw puzzle, pieces are connected horizontally and vertically by a connector--an "out" fitting in an "in". For each connection there are two possible choices: the "out" connector can point left or right, up or down. Assume that this choice is made entirely at random for each connector.
In the puzzle box we find various shapes of pieces. Here are some examples:(Note: the orientation of a shape is not relevant. For instance, the second drawing represents all pieces with three "in" and one "out".)
What is the expected ratio, \(p:q:r:s\), of the number of pieces of each shape?