In how many ways can the 2 x 10 board shown below be covered with 1 x 2 dominos?

[BKelAB]

In how many ways can the 2 x 10 board shown below be covered with 1 x 2 dominos?

[Soroban]

Hello, BKelAB!

I think I have an approach,
. . but it requires a bit of Listing.

There are two types of dominos:

. . Horizontal

. . . . Vertical


Note that two 's form a 2-by-2 square.

. . Let represent: .



There are 6 cases to consider:


(1) 10 V's: there is 1 way.


(2) 8 V's, 1 HH

. . .Place the 8 V's in a row, leaving spaces before, after and between them.
. . . .

. . .The can be placed in any of the 9 spaces.

. . There are 9 ways with 8 V's and 1 HH.


(3) 6 V's, 2 HH's

. . .Place the 6 V's in a row, leaving spaces before, after and between them.
. . . .

. . .Each HH has 7 choices of spaces: . choices.

. . .There are 49 ways with 6 V's and 2 HH's.


(4) 4 V's, 3 HH's

. . .Place the 4 V's in a row, leaving spaces before, after and between them.
. . . .

. . .Each HH has 5 choices of spaces: . choices.

. . .There are 125 ways with 4 V's and 3 HH's.


(5) 2V's, 4HH's

. . .Place the 4 HH's in a row, leaving spaces before, after and between them.

. . . .

. . .Each V has 5 choices for spaces.: . choices.

. . .There are 25 ways with 2V's and 4 HH's.


(6) 5 HH's: There is 1 way.



Therefore, there are: . ways.