# A parallelogram is cut by two sets of m lines parallel to its sides. Find the number of parallelograms thus formed.

We know that parallelogram has 4 lines in which 2 sides are parallel to each other which means 2 pairs of lines are parallel lines

It is told that parallelogram is cut by two sets of m lines parallel to its sides.

This means there will be two sets of (m + 2) lines parallel to each other.

We need two sets of parallel lines to form a parallelogram in which the lines need to be chosen from these two sets of (m + 2) parallel lines.

Let us assume that the no. of parallelograms formed be N.

N = (choosing 2 parallel lines from (m + 2) parallel lines which are parallel to one side) × (choosing 2 parallel lines from (m + 2) parallel lines which are parallel to the side which is not parallel to the first side)

N = (m + 2C2) × (m + 2C2)

N = (m + 2C2)2

We know that ,

And also n! = (n)(n – 1)......2.1    The total no. of parallelograms formed are .

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