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.
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III Six dominoes vertical (4 horizontal)
The result of this is 35 ways to arrange the dominoes.
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IV Four dominoes vertical (6 horizontal)
The result of this is 28 ways to arrange the dominoes.
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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!