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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
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

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imqwerty
 one year ago
Best ResponseYou've already chosen the best response.1how many squares how many parallelograms

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1dw:1441070677769:dw

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1\(\large \color{black}{\begin{align} & \normalsize \text{Each of 8 identical balls is to be placed in the squares shown} \hspace{.33em}\\~\\ & \normalsize \text{ in the figure given in a horizontal direction such that one } \hspace{.33em}\\~\\ & \normalsize \text{horizontal row contains 6 balls and the other horizontal row} \hspace{.33em}\\~\\ & \normalsize \text{ contains 2 balls .In how many different ways can this be done} \hspace{.33em}\\~\\ & a.)\ 38 \hspace{.33em}\\~\\ & a.)\ 28 \hspace{.33em}\\~\\ & a.)\ 16 \hspace{.33em}\\~\\ & a.)\ 14 \hspace{.33em}\\~\\ \end{align}}\)

dan815
 one year ago
Best ResponseYou've already chosen the best response.1so for 1 complete horizontal row filled we have 6Choose 2 + 4so 2* (6 choose 2 + 4) for the complete case?

imqwerty
 one year ago
Best ResponseYou've already chosen the best response.1dw:1441070953833:dw

dan815
 one year ago
Best ResponseYou've already chosen the best response.1im not sure if that is possible because then the 6 are vertical

dan815
 one year ago
Best ResponseYou've already chosen the best response.1that why in the beginning they were like oh its horizontal given this picture as it is

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4dan's \(\large \dbinom{2}{1}*(4+\dbinom{6}{2})\) looks neat to me!

dan815
 one year ago
Best ResponseYou've already chosen the best response.1plus 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
 one year ago
Best ResponseYou've already chosen the best response.1keeping ball B still at that position and moving ball A anywhere between 16x6 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 16 with ball A so get 6x5 more cases =30 so the answer =24+30=54

mathmate
 one year ago
Best ResponseYou've already chosen the best response.1With horizontal only, I also have \(\Large 2*\dbinom{6}{6}*(\dbinom{6}{2}+4\dbinom{2}{2})=38\)

imqwerty
 one year ago
Best ResponseYou've already chosen the best response.1did i do something wrong?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4``` keeping ball B still at that position and moving ball A anywhere between 16x6 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
 one year ago
Best ResponseYou've already chosen the best response.4the 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
 one year ago
Best ResponseYou've already chosen the best response.4``` keeping ball B still at that position and moving ball A anywhere between 16x6 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 16 with ball A so get 6x5 more cases =30 ``` simply divide that by 2 to fix that duplicates issue

mathmate
 one year ago
Best ResponseYou've already chosen the best response.1@dan815 Have you worked out the number of rectangles yet? lol

imqwerty
 one year ago
Best ResponseYou've already chosen the best response.1dw:1441072190782:dw

imqwerty
 one year ago
Best ResponseYou've already chosen the best response.1in this case where i m moving ball A only in 16 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
 one year ago
Best ResponseYou've already chosen the best response.4no there are only 8 balls you must put 2 balls horizontal in one row and 6 balls horizontal in another row

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I 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
 one year ago
Best ResponseYou've already chosen the best response.1hey since the asker is gone i thought wed go over htis problem too, so how many rectangles/squares are there in that figure

dan815
 one year ago
Best ResponseYou've already chosen the best response.1i tried 2*(3choose 2 * 7 choose 2)  (3 choose 2)^2

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.429 squares easy to count one by one :)

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.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

dan815
 one year ago
Best ResponseYou've already chosen the best response.1right thats what i was thinking

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4right im just translating ur math symbols to layman english

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4Here is a more complicated but fun problem find the order of the group of symmetries of given pattern

dan815
 one year ago
Best ResponseYou've already chosen the best response.1like the number of every unique rectangle?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4a group is a set with a binary operation that is associative, has an identity element and inverses exist for each element

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4here the group is the set of all symmetries of given pattern operation is "composition"

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4by "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
 one year ago
Best ResponseYou've already chosen the best response.4similarly 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
 one year ago
Best ResponseYou've already chosen the best response.1i didn't understand this \(\large \dbinom{2}{1}\times \left(4+\dbinom{6}{2}\right)\)

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.1sry no teamviewer , m on mobile connection

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4its 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
 one year ago
Best ResponseYou've already chosen the best response.4look at the given pattern how many rows have at least 6 squares ?

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.41) `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
 one year ago
Best ResponseYou've already chosen the best response.41) 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

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4let me know once you digest step1

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4dw:1441074443487:dw

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4after step1, situation might look something like above

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.42) 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

ganeshie8
 one year ago
Best ResponseYou've already chosen the best response.4dw:1441074636585:dw