by Frater Elijah
Suppose that we want to pave the bottom of a box of size 2 x 10 with 1 x 2 dominoes. In how many ways can this be done?
We shall accomplish this problem using the factorial representation for picking k objects from n objects. Since we want to tile this 2 x 10 box with 1 x 2 dominoes we must consider all "vertical" dominoes in groups of two, while all horizontal dominoes can be considered as a single group unto themselves. Consult the following cases:
I All dominoes vertical
The result of this is 1, way to do this.
II Eight dominoes vertical (2 horizontal)
The result of this is 15, ways to do this.
III Six dominoes vertical (4 horizontal)
The result of this is 35 ways to arrange the dominoes.
IV Four dominoes vertical (6 horizontal)
The result of this is 28 ways to arrange the dominoes.
V Two dominoes vertical (8 horizontal)
The result of this is 9 ways to arrange the dominoes.
VI 0 dominoes vertical (10 horizontal)
The result of this is 1 way to arrange all of these dominoes.
Now taking the sum of all these individual cases yields the total number of ways to arrange the dominoes:1+15+35+28+9+1= 89 different ways!