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

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

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

2. Hello, BKelAB!

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

There are two types of dominos:

. . Horizontal $(H)\!:\;\;\sqsubset \!\sqsupset$

. . . . Vertical $(V)\!:\;\;\begin{array}{c} \sqcap \\ [-3mm] \sqcup \end{array}$

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

. . Let $H\!H$ represent: . $\begin{array}{c}\sqsubset\! \sqsupset \\ [-3mm] \sqsubset\! \sqsupset \end{array}$

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.
. . . . $\_ \;V\;\_\;V\;\_\;V\;\_\;V\;\_\;V\;\_\; V\;\_\;V\;\_\;V\;\_$

. . .The $H\!H$ 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.
. . . . $\_\;V\;\_\;V\;\_\;V\;\_\:V\;\_\;V\;\_\;V\;\_$

. . .Each HH has 7 choices of spaces: . $7^2$ 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.
. . . . $\_\;V\;\_\;V\;\_\;V\;\_\;V\;\_$

. . .Each HH has 5 choices of spaces: . $5^3$ 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.

. . . . $\_\;H\!H\;\_\;H\!H\;\_\;H\!H\;\_\;H\!H \;\_$

. . .Each V has 5 choices for spaces.: . $5^2$ choices.

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

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

Therefore, there are: . $1 + 9 + 49 + 125 + 25 + 1 \;=\;{\color{blue}210}$ ways.