A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

mathmath333

  • one year ago

Each of 8 identical balls is to be placed in the squares shown in the figure given in a horizontal direction such that one horizontal row contains 6 balls and the other horizontal row contains 2 balls .In how many different ways can this be done

  • This Question is Closed
  1. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    1 Attachment
  2. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    i am dizzy already

  3. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    how many squares how many parallelograms

  4. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1441070677769:dw|

  5. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    \(\large \color{black}{\begin{align} & \normalsize \text{Each of 8 identical balls is to be placed in the squares shown} \hspace{.33em}\\~\\ & \normalsize \text{ in the figure given in a horizontal direction such that one } \hspace{.33em}\\~\\ & \normalsize \text{horizontal row contains 6 balls and the other horizontal row} \hspace{.33em}\\~\\ & \normalsize \text{ contains 2 balls .In how many different ways can this be done} \hspace{.33em}\\~\\ & a.)\ 38 \hspace{.33em}\\~\\ & a.)\ 28 \hspace{.33em}\\~\\ & a.)\ 16 \hspace{.33em}\\~\\ & a.)\ 14 \hspace{.33em}\\~\\ \end{align}}\)

  6. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    none of the above

  7. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so for 1 complete horizontal row filled we have 6Choose 2 + 4so 2* (6 choose 2 + 4) for the complete case?

  8. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    38

  9. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1441071022533:dw|

  10. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1441070953833:dw|

  11. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    im not sure if that is possible because then the 6 are vertical

  12. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    that why in the beginning they were like oh its horizontal given this picture as it is

  13. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    dan's \(\large \dbinom{2}{1}*(4+\dbinom{6}{2})\) looks neat to me!

  14. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    plus if it is vertical u can no longer have a case where the remaining 2 can be a pair of horizontal theyd have to be tripple or 2 pairs of horizontal

  15. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    keeping ball B still at that position and moving ball A anywhere between 1-6x6 we get 6 cases ball B can go anywhere between its original postion -13 so we get 6x4 cases =24 ball B can also be put at any position 1-6 with ball A so get -6x5 more cases =30 so the answer =24+30=54

  16. mathmate
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    With horizontal only, I also have \(\Large 2*\dbinom{6}{6}*(\dbinom{6}{2}+4\dbinom{2}{2})=38\)

  17. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    did i do something wrong?

  18. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    ``` keeping ball B still at that position and moving ball A anywhere between 1-6x6 we get 6 cases ball B can go anywhere between its original postion -13 so we get 6x4 cases =24 ``` Nice idea, but there are some duplicates, double check

  19. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    the duplicates are because of counting AB and BA as two different things but the balls are identical, so AB and BA must be counted only once |dw:1441071598536:dw|

  20. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    ``` keeping ball B still at that position and moving ball A anywhere between 1-6x6 we get 6 cases ball B can go anywhere between its original postion -13 so we get 6x4 cases =24 ball B can also be put at any position 1-6 with ball A so get -6x5 more cases =30 ``` simply divide that by 2 to fix that duplicates issue

  21. mathmate
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @dan815 Have you worked out the number of rectangles yet? lol

  22. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    haha xD

  23. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1441072190782:dw|

  24. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    in this case where i m moving ball A only in 1-6 we r getting 13x6 cases woah its even more nd we can assume the question to be this - find the number of possible combinations ...keeping 6 balls in a row with 2 balls free to move its ok to no to consider that 2balls kept in horizontal cause that will happen in each and every case

  25. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    no there are only 8 balls you must put 2 balls horizontal in one row and 6 balls horizontal in another row

  26. anonymous
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I said 64 because 8 goes into 6 = 48 time and 8 goes into 2 = 16 time then i added them together and got 64

  27. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hey since the asker is gone i thought wed go over htis problem too, so how many rectangles/squares are there in that figure

  28. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1441072848851:dw|

  29. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i tried 2*(3choose 2 * 7 choose 2) - (3 choose 2)^2

  30. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    29 squares easy to count one by one :)

  31. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1441072894766:dw|

  32. imqwerty
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    thanks @ganeshie8 :D

  33. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    ` 2*(3choose 2 * 7 choose 2) - (3 choose 2)^2` looks good to me basically you need 2 horizontal lines and 2 vertical lines to form a rectangle

  34. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    right thats what i was thinking

  35. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    right im just translating ur math symbols to layman english

  36. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    Here is a more complicated but fun problem find the order of the group of symmetries of given pattern

  37. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    o boy

  38. dan815
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    like the number of every unique rectangle?

  39. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    a group is a set with a binary operation that is associative, has an identity element and inverses exist for each element

  40. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    here the group is the set of all symmetries of given pattern operation is "composition"

  41. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    by "set of all symmetries" i mean all the transformations that bring the pattern back to itself for example, rotating the pattern by 90 degrees wont change it so \(R_{90}\) is a group element

  42. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    similarly a horizontal flip wont change the pattern so "horizontal flip" is also a group element i think it is an easy problem, its just the terminology that gets in the way..

  43. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i didn't understand this \(\large \dbinom{2}{1}\times \left(4+\dbinom{6}{2}\right)\)

  44. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    teamviewer ?

  45. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    wait

  46. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    sry no teamviewer , m on mobile connection

  47. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    2 rows

  48. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    its okay so we are doing the problem in two steps : 1) choose a row, then choose 6 squares in that row for placing 6 balls 2) after that, choose another row and 2 squares in that row for placing 2 balls

  49. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    look at the given pattern how many rows have at least 6 squares ?

  50. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    2 rows ?

  51. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    1) `choose a row`, then choose 6 squares in that row for placing 6 balls yes, you can choose \(1\) row from the available \(2\) rows in \(\dbinom{2}{1}\) ways

  52. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    1) choose a row, then `choose 6 squares in that row for placing 6 balls` then you can choose 6 squares in that row in \(\dbinom{6}{6}\) ways so step1 can be done in \(\dbinom{2}{1}*\dbinom{6}{6}\) ways

  53. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ok

  54. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    let me know once you digest step1

  55. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    i digested

  56. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    |dw:1441074443487:dw|

  57. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    after step1, situation might look something like above

  58. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    lets see step2

  59. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes

  60. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    2) after that, `choose another row` and 2 squares in that row for placing 2 balls here we need to be careful because not all rows have same number of squares

  61. mathmath333
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    2*6C2+4*2C2

  62. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 4

    |dw:1441074636585:dw|