Hello, BKelAB!

II have an approach,think

. . 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.

. . . .

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

. . .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.

. . . .

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

. . .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.

. . . .

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

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

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

Therefore, there are: . ways.