parallel lines

**arunachalam.s** · August 18th 2009, 11:09 PM

If a set of 10 parallel lines are intersected by a set of 13 other parallel lines,then find the number of parallelograms formed?

I m not able to undertsand the question ..is there a standard formula for this type of questions

**red_dog** · August 18th 2009, 11:52 PM

Let $S_1, \ S_2$ be the two sets of parallel lines. A paralleogram is formed by a pair of lines from $S_1$ and a pair of lines from $S_2$ . So, the number of parallelograms depends on how many pairs can take from $S_1$ and $S_2$ .

This number is $C_{10}^2\cdot C_{13}^2=45\cdot 78=3510$

Generally, if there is a set of n parallel lines and another set of m parallel lines, the number of parallelograms is $C_n^2\cdot C_m^2$

**arunachalam.s** · August 19th 2009, 01:08 AM

will this formula hold good if both m and n are equal like in a chess board 9 parallel lines intersecting 9 parallel lines so if i apply ur formula 9C2*9C2 is not equal to 64 .

so i did like this number of lines -1C (min number of lines -1)
so 22-1C10-1=21C9

**Plato** · August 19th 2009, 03:19 AM

Originally Posted by arunachalam.s

will this formula hold good if both m and n are equal like in a chess board 9 parallel lines intersecting 9 parallel lines so if i apply ur formula 9C2*9C2 is not equal to 64 .

You have a missunderstaning about the count.
Look at the attached 3x3 grid.
There are a total of nine rectangles in that grid: $\binom{3}{2}\binom{3}{2}=9$ .

**arunachalam.s** · August 19th 2009, 04:32 AM

k i understand thank u

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