OH!.......Parellelograms

[Phlembac Adib Hasan]

( Inspired from Masum Vaia ) In co-ordinate plane, join points

$[(0,0),(m,0)],[(0,1),(m,1)],...,[(o,n),(m,n)]$ and

$[(0,0],(0,n)], [(1,0),(1,n)...,[(m,0),(m,n)]$.How many parallelograms can you make

joining the intersection points of these lines?


Note:$(0,0),(2,1),(3,3),(1,2)$ is also a parallelogram.So you must count this type of parallelograms too!

[nafistiham]

let me share a hint given by the topic owner [without permission ]

a parallelogram needs just two pairs of parallel lines, try combinatorics.

[the topic owner does not want you to read it ]

[Phlembac Adib Hasan]

Not Fare.

[nafistiham]

Not Fare.

ok.i am hiding that.

[Phlembac Adib Hasan]

Ok.Ok.It's alright now.

[nafistiham]

my solution was like this
\[\sum_{k=1}^{n-1}k \cdot\sum_{p=1}^{m-1}p\]

adib made it short

[Phlembac Adib Hasan]

my solution was like this
\[\sum_{k=1}^{n-1}k \cdot\sum_{p=1}^{m-1}p\]

adib made it short

You might do a mistake.Because you are not counting $(0,0),(2,1),(3,3),(1,2)$ type parallelograms,only counting the rectangles.