This the problem from the test and I'm stuck at one part.
Matt will arrange four identical, dotless dominoes (shaded 1 by 2
rectangles) on the 5 by 4 grid to the right so that a path is formed
from the upper left-hand corner A to the lower right hand corner B.
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?
So do I just find all the numbers of vertical and horizontal possible and then multiply by the number of arrangements of each, or should I go a different route?
Best Answer
You need to traverse $7$ cells (you start from $A$, so that cell is a given). You need to make exactly $3$ movements rightwards in total (otherwise we won't reach point $B$'s x-coordinate).
So we are looking for the number of ways we can order $$rrrdddd$$ where $r$ denotes going right, and $d$ denotes going down.
That gives $$\binom 73 = 35$$ ways.