## 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

1. mathmath333

2. anonymous

3. imqwerty

how many squares how many parallelograms

4. mathmath333

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5. mathmath333

\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

none of the above

7. dan815

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

8. dan815

38

9. dan815

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10. imqwerty

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11. dan815

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

12. dan815

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

13. ganeshie8

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

14. dan815

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

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

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

17. imqwerty

did i do something wrong?

18. ganeshie8

 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

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

 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

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

22. dan815

haha xD

23. imqwerty

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24. imqwerty

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

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

26. anonymous

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

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

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29. dan815

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

30. ganeshie8

29 squares easy to count one by one :)

31. dan815

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32. imqwerty

thanks @ganeshie8 :D

33. ganeshie8

 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

right thats what i was thinking

35. ganeshie8

right im just translating ur math symbols to layman english

36. ganeshie8

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

37. dan815

o boy

38. dan815

like the number of every unique rectangle?

39. ganeshie8

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

40. ganeshie8

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

41. ganeshie8

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

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

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

44. ganeshie8

teamviewer ?

45. mathmath333

wait

46. mathmath333

sry no teamviewer , m on mobile connection

47. mathmath333

2 rows

48. ganeshie8

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

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

50. mathmath333

2 rows ?

51. ganeshie8

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

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

ok

54. ganeshie8

let me know once you digest step1

55. mathmath333

i digested

56. ganeshie8

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57. ganeshie8

after step1, situation might look something like above

58. ganeshie8

lets see step2

59. mathmath333

yes

60. ganeshie8

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

2*6C2+4*2C2

62. ganeshie8

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