mathmath333
  • mathmath333
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
Mathematics
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schrodinger
  • schrodinger
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mathmath333
  • mathmath333
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anonymous
  • anonymous
i am dizzy already
imqwerty
  • imqwerty
how many squares how many parallelograms

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mathmath333
  • mathmath333
|dw:1441070677769:dw|
mathmath333
  • mathmath333
\(\large \color{black}{\begin{align} & \normalsize \text{Each of 8 identical balls is to be placed in the squares shown} \hspace{.33em}\\~\\ & \normalsize \text{ in the figure given in a horizontal direction such that one } \hspace{.33em}\\~\\ & \normalsize \text{horizontal row contains 6 balls and the other horizontal row} \hspace{.33em}\\~\\ & \normalsize \text{ contains 2 balls .In how many different ways can this be done} \hspace{.33em}\\~\\ & a.)\ 38 \hspace{.33em}\\~\\ & a.)\ 28 \hspace{.33em}\\~\\ & a.)\ 16 \hspace{.33em}\\~\\ & a.)\ 14 \hspace{.33em}\\~\\ \end{align}}\)
imqwerty
  • imqwerty
none of the above
dan815
  • dan815
so for 1 complete horizontal row filled we have 6Choose 2 + 4so 2* (6 choose 2 + 4) for the complete case?
dan815
  • dan815
38
dan815
  • dan815
|dw:1441071022533:dw|
imqwerty
  • imqwerty
|dw:1441070953833:dw|
dan815
  • dan815
im not sure if that is possible because then the 6 are vertical
dan815
  • dan815
that why in the beginning they were like oh its horizontal given this picture as it is
ganeshie8
  • ganeshie8
dan's \(\large \dbinom{2}{1}*(4+\dbinom{6}{2})\) looks neat to me!
dan815
  • 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
imqwerty
  • 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
mathmate
  • mathmate
With horizontal only, I also have \(\Large 2*\dbinom{6}{6}*(\dbinom{6}{2}+4\dbinom{2}{2})=38\)
imqwerty
  • imqwerty
did i do something wrong?
ganeshie8
  • 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
ganeshie8
  • 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|
ganeshie8
  • 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
mathmate
  • mathmate
@dan815 Have you worked out the number of rectangles yet? lol
dan815
  • dan815
haha xD
imqwerty
  • imqwerty
|dw:1441072190782:dw|
imqwerty
  • 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
ganeshie8
  • ganeshie8
no there are only 8 balls you must put 2 balls horizontal in one row and 6 balls horizontal in another row
anonymous
  • 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
dan815
  • 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
dan815
  • dan815
|dw:1441072848851:dw|
dan815
  • dan815
i tried 2*(3choose 2 * 7 choose 2) - (3 choose 2)^2
ganeshie8
  • ganeshie8
29 squares easy to count one by one :)
dan815
  • dan815
|dw:1441072894766:dw|
imqwerty
  • imqwerty
thanks @ganeshie8 :D
ganeshie8
  • 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
dan815
  • dan815
right thats what i was thinking
ganeshie8
  • ganeshie8
right im just translating ur math symbols to layman english
ganeshie8
  • ganeshie8
Here is a more complicated but fun problem find the order of the group of symmetries of given pattern
dan815
  • dan815
o boy
dan815
  • dan815
like the number of every unique rectangle?
ganeshie8
  • ganeshie8
a group is a set with a binary operation that is associative, has an identity element and inverses exist for each element
ganeshie8
  • ganeshie8
here the group is the set of all symmetries of given pattern operation is "composition"
ganeshie8
  • 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
ganeshie8
  • 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..
mathmath333
  • mathmath333
i didn't understand this \(\large \dbinom{2}{1}\times \left(4+\dbinom{6}{2}\right)\)
ganeshie8
  • ganeshie8
teamviewer ?
mathmath333
  • mathmath333
wait
mathmath333
  • mathmath333
sry no teamviewer , m on mobile connection
mathmath333
  • mathmath333
2 rows
ganeshie8
  • 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
ganeshie8
  • ganeshie8
look at the given pattern how many rows have at least 6 squares ?
mathmath333
  • mathmath333
2 rows ?
ganeshie8
  • 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
ganeshie8
  • 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
mathmath333
  • mathmath333
ok
ganeshie8
  • ganeshie8
let me know once you digest step1
mathmath333
  • mathmath333
i digested
ganeshie8
  • ganeshie8
|dw:1441074443487:dw|
ganeshie8
  • ganeshie8
after step1, situation might look something like above
ganeshie8
  • ganeshie8
lets see step2
mathmath333
  • mathmath333
yes
ganeshie8
  • 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
mathmath333
  • mathmath333
2*6C2+4*2C2
ganeshie8
  • ganeshie8
|dw:1441074636585:dw|