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

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- mathmath333

##### 1 Attachment

- anonymous

i am dizzy already

- imqwerty

how many squares
how many parallelograms

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## More answers

- mathmath333

|dw:1441070677769:dw|

- 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

none of the above

- dan815

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

- dan815

38

- dan815

|dw:1441071022533:dw|

- imqwerty

|dw:1441070953833:dw|

- dan815

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

- dan815

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

- ganeshie8

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

- 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

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

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

- imqwerty

did i do something wrong?

- 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

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

```
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

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

- dan815

haha xD

- imqwerty

|dw:1441072190782:dw|

- 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

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

- 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

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

|dw:1441072848851:dw|

- dan815

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

- ganeshie8

29 squares
easy to count one by one :)

- dan815

|dw:1441072894766:dw|

- imqwerty

thanks @ganeshie8 :D

- 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

right thats what i was thinking

- ganeshie8

right im just translating ur math symbols to layman english

- ganeshie8

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

- dan815

o boy

- dan815

like the number of every unique rectangle?

- ganeshie8

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

- ganeshie8

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

- 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

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

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

- ganeshie8

teamviewer ?

- mathmath333

wait

- mathmath333

sry no teamviewer , m on mobile connection

- mathmath333

2 rows

- 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

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

- mathmath333

2 rows ?

- 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

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

ok

- ganeshie8

let me know once you digest step1

- mathmath333

i digested

- ganeshie8

|dw:1441074443487:dw|

- ganeshie8

after step1, situation might look something like above

- ganeshie8

lets see step2

- mathmath333

yes

- 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

2*6C2+4*2C2

- ganeshie8

|dw:1441074636585:dw|