Results 1 to 2 of 2

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

  1. #1
    Newbie
    Joined
    Oct 2009
    Posts
    7

    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?

    Last edited by mr fantastic; February 8th 2010 at 01:10 AM. Reason: Restored deleted question
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,824
    Thanks
    713
    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.

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Sorting by Covered-Rest Distance and Time
    Posted in the Statistics Forum
    Replies: 0
    Last Post: October 15th 2011, 12:56 PM
  2. Replies: 1
    Last Post: December 14th 2009, 06:59 PM
  3. calculation of Effective area covered
    Posted in the Geometry Forum
    Replies: 2
    Last Post: January 25th 2009, 05:40 PM
  4. I m taking calc 2 (topics covered)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 17th 2008, 02:25 PM
  5. Replies: 5
    Last Post: March 25th 2008, 09:52 PM

Search Tags


/mathhelpforum @mathhelpforum