Browse · MATH
Printjmc
counting and probability senior
Problem
Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid below so that a path is formed from the upper left-hand corner to the lower righthand corner . In a path, consecutive dominoes must touch at their sides and not just their corners. No domino may be placed diagonally; each domino covers exactly two of the unit squares shown on the grid. One arrangement is shown. How many distinct arrangements are possible, including the one shown?
Solution
The shortest possible path from to requires dominoes, which is all we have, so we must use them to make only down and right movements - we have none to waste going up or left. We need to make movements to the right and down, and we can arrange them however we wish. So there are
arrangements.
It is easy to see that each domino arrangement is one of the path mentioned above. To show every above mentioned path can be paved by the dominoes, color the table cells white and black alternatively. Then each path must also be white and black alternatively, thus can always be paved by the dominoes.
arrangements.
It is easy to see that each domino arrangement is one of the path mentioned above. To show every above mentioned path can be paved by the dominoes, color the table cells white and black alternatively. Then each path must also be white and black alternatively, thus can always be paved by the dominoes.
Final answer
35